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\begin{document}

\title[Kodaira dimension]{The Kodaira dimension of spaces of rational
curves on low degree hypersurfaces}

\author[Starr]{Jason Michael Starr}
\address{Department of Mathematics \\
  Massachusetts Institute of Technology \\ Cambridge MA 02139}
\email{jstarr@math.mit.edu} 
\date{\today}


\begin{abstract}
For a hypersurface in complex projective space, $X\subset \PP^n$, we
investigate the singularities and Kodaira dimension of the Kontsevich
moduli spaces $\Kbm{0,0}{X,e}$ parametrizing rational curves of degree
$e$ on $X$.  If $d+e \leq n$ and $X$ is a general hypersurface of
degree $d$, we prove that $\Kbm{0,0}{X,e}$ has only canonical
singularities and we conjecture the same is true for the coarse moduli
space $\kbm{0,0}{X,e}$.  We prove that this conjecture is implied by
the ``inversion of adjunction'' conjecture of Koll\'{a}r and Shokurov.
Also we compute the canonical divisor of $\Kbm{0,0}{X,e}$ and show
that for most pairs $(d,e)$ with $n \leq d^2 \leq n^2$, the canonical
divisor is a \emph{big} divisor.  When combined with the above
conjecture, this implies that in many cases $\kbm{0,0}{X,e}$ is a
variety of general type.  This investigation is motivated by the
question of which Fano hypersurfaces are unirational.
\end{abstract}

\maketitle


\section{Introduction}~\label{intro}

\medskip\noindent 
Let $X\subset \PP^n$ be a general hypersurface of degree $d$ in
complex projective space $\PP^n$.  The Kontsevich moduli space
$\Kbm{0,0}{X,e}$ is a proper, Deligne-Mumford stack containing as
an open substack the scheme parametrizing smooth, rational curves of
degree $e$ on $X$ (c.f. ~\cite{FP}).  Except when $d=1$, $d=2$ or
$e=1$, very little is known about the singularities and Kodaira
dimension of $\Kbm{0,0}{X,e}$.  When $d=1$ and $d=2$, the spaces
$\Kbm{0,0}{X,e}$ are all smooth ~\cite{Kv} and rational ~\cite{KP}.
When $e=1$, the space $\Kbm{0,0}{X,e}$ is just the space of lines on
$X$, which is completely understood ~\cite[Thm. V.4.3]{K}.

\medskip\noindent 
What is known for $d \geq 3$ and $e \geq 2$?  If $d < \frac{n+1}{2}$,
it is proved in ~\cite{HRS2} that $\Kbm{0,0}{X,e}$ is integral of the
expected dimension and has only local complete intersection
singularities.  If also $d^2 + d +1 < n$, it is proved in ~\cite{HS2}
that $\Kbm{0,0}{X,e}$ has negative Kodaira dimension;  in fact
$\Kbm{0,0}{X,e}$ is \emph{rationally connected}.  If $d \geq n-1$, the
open substack of $\Kbm{0,0}{X,e}$ parametrizing smooth rational curves
is not Zariski dense: the locus of \emph{multiple covers of lines}
yields an irreducible component of $\Kbm{0,0}{X,e}$ not contained in
the closure of this open set.  However all evidence suggests that for
$d\leq n-2$ and for $X\subset \PP^n$ a general hypersurface of degree
$d$, the stack $\Kbm{0,0}{X,e}$ is irreducible for all $e\geq 1$.

\medskip\noindent
\begin{ques} \label{ques-0} 
For $d \leq n-2$ and $X\subset \PP^n$ a general hypersurface of degree
$d$, what type of singularities does $\Kbm{0,0}{X,e}$ have?  For which
$(n,d,e)$ are the singularities terminal, resp. canonical, log
canonical?
\end{ques}

\medskip\noindent 
Fix $e > 1$ and $d\geq 1$.  Let $\PP^N$ denote the projective space
parametrizing degree $d$ hypersurfaces $X\subset \PP^n$.  Let $C_d
\subset \PP^N \times \Kbm{0,0}{\PP^n,e}$ denote the closed substack
parametrizing pairs $([X],[f:D \rightarrow X])$ of $X\subset \PP^n$
a hypersurface and $f:D \rightarrow X$ a stable map in
$\Kbm{0,0}{X,e}$.  The main theorem of this paper is the following.

\medskip\noindent
\setcounter{section}{7}
\setcounter{theorem}{4}
\begin{thm} 
If $e\geq 2$ and if $d+e \leq n$, then $C_d$ is an integral, normal,
local complete intersection stack of the expected dimension and has at worst
canonical singularities.
\end{thm}

\medskip\noindent
\setcounter{section}{7}
\setcounter{theorem}{5}
\begin{cor} 
If $e \geq 2$ and if $d+e \leq n$, then for a general hypersurface
$X\subset \PP(V)$ of degree $d$, the Kontsevich moduli space
$\Kbm{0,0}{X,e}$ is an integral, normal, local complete intersection
stack of the expected dimension $(n+1-d)e+(n-3)$ and has at worst canonical
singularities.
\end{cor}
\setcounter{section}{1}
\setcounter{theorem}{1}

\medskip\noindent 
It is reasonable to expect that Theorem~\ref{thm-main} is sharp.
For instance the inequality $d+e \leq n$ is consistent with the
inequality $d \leq n-1$ necessary for $C_d$ to be irreducible.  On the
other hand, Corollary~\ref{cor-main} is certainly not sharp: it fails
to account for the cases $d=1$ and $d=2$ where $\Kbm{0,0}{X,e}$ is
smooth for every $e$.  For $d \geq 3$ and $e\geq 2$, the space
$\Kbm{0,0}{X,e}$ \emph{is singular}, but hopefully it is not too
singular.

\medskip\noindent
\begin{ques} \label{ques-1} 
For which integers $d$ and $n$ is it true that for a very general
hypersurface $X\subset \PP^n$ of degree $d$, $X$ is unirational?
\end{ques}

\medskip\noindent 
This is the question which motivates this paper, although no new
answers are given here.  A necessary condition is that $d \leq n$;
otherwise the Kodaira dimension of $X$ is nonnegative.  For $d=1,2,3$ and
$n \geq d$, it is known that a general hypersurface $X \subset \PP^n$
of degree $d$ is unirational.  For each integer $d$ there is an
integer $\phi(d)$ such that for $n \geq \phi(d)$ and $X\subset \PP^n$
a general hypersurface, then $X$ is unirational (~\cite{HMP},
~\cite[Chapter 23]{K+92}).  It is conjectured that if $d \leq n$ but
large compared to $n$ -- e.g. if $d=n$ for $n \geq 4$ -- then a
general hypersurface $X\subset \PP^n$ is \emph{not} unirational.  But
no example of such a hypersurface has been proved to be
non-unirational.

\subsection{Koll\'{a}r's approach}
\medskip\noindent 
The connection between Question~\ref{ques-1} and this paper comes from
a suggestion by Koll\'{a}r in ~\cite{KollarSimple}, that a necessary
condition for a variety $X$ of dimension at least $2$ to be
unirational is that for a general point $p\in X$, there exists a
rational surface $S\subset X$ containing $p$.

\medskip\noindent
\begin{defn} \label{defn-manyRS} 
An irreducible, projective variety $X$ over a field $k$ is \emph{swept
by rational surfaces} (resp. \emph{separably swept by rational
surfaces}) if there exists an irreducible variety $Z$ and a rational
transformation $F: Z \times \PP^2 \rightarrow X$ such that,
\begin{enumerate}
\item[(i)] 
the rational transformation $F$ is dominant (resp. dominant
and separable), and
\item[(ii)] 
the rational transformation $(\text{pr}_Z,F): Z \times \PP^2
\rightarrow Z \times X$ is generically finite to its image
(resp. generically finite and separable to its image).
\end{enumerate}
\end{defn}

\medskip\noindent
\begin{rmk} \label{rmk-manyRS} 
There are several obvious remarks.
\begin{enumerate}
\item[(i)] 
This definition makes sense for any field $k$, not necessarily
algebraically closed nor of characteristic $0$ (although this is
the case of interest in the rest of the paper).
\item[(ii)] 
In the definition above, $Z$ can be replaced by a separable, dominant
cover and the conditions will still hold.
\item[(iii)] 
Let $X$ be a variety that is separably swept by rational surfaces.
Let $S \subset Z \times X$ denote the image of $(\text{pr}_Z,F)$.  By
~\cite[Thm. III.2.4]{K}, the base change $S \otimes_{K(Z)}
\overline{K}(Z)$ is a rational surface.  By ~\cite{Coombes}, in fact
there is a separable dominant morphism $Z'\rightarrow Z$ and a
birational transformation over $Z'$, $G: Z' \times \PP^2 \rightarrow
Z' \times_Z S$.  After replacing $Z$ by $Z'$ and replacing $F$ by the
composite $Z' \times \PP^2 \xrightarrow{G} Z'\times_Z S
\xrightarrow{\text{pr}_S} S \xrightarrow{\text{pr}_X} X$, 
$(\text{pr}_Z,F)$ is actually birational to its image.
\item[(iv)] 
If $X$ is (separably) unirational and $\text{dim}(X) \geq 2$, then $X$
is (separably) swept by rational surfaces.
\item[(v)] 
If $X$ is swept by rational surfaces (resp. separably swept by
rational surfaces), then $X$ is uniruled (resp. separably uniruled).
\item[(vi)] 
Let $\text{dim}(X) = n$.  Definition~\ref{defn-manyRS} is equivalent to the
stronger condition where $\text{dim}(Z) = n-2$.
Moreover $Z$ can be required to be smooth over $k$.  In fact, by de
Jong's alterations of singularities, up to replacing $Z$ by a
generically \'etale cover, $Z$ can even be required to be smooth and
projective.
\item[(vii)] 
The condition that $X$ is swept by rational surfaces (resp. separably
swept by rational surfaces) is a birational property that is
equivalent to the condition that there exist a finitely-generated
field extension $L/k$ of transcendence degree $n-2$ and a finite
super-extension (resp. finite separable super-extension) of $K(X)/k$
of the form $L(t_1,t_2)/K(X)$ (resp. such that the compositum $L*K(X)$
equals $L(t_1,t_2)$).
\item[(viii)] 
If $X$ is swept by rational surfaces (resp. separably swept by
rational surfaces) and $f:X \rightarrow X'$ is a generically finite,
dominant rational transformation (resp. generically \'etale, dominant
rational transformation), then also $X'$ is swept by rational surfaces
(resp. separably swept by rational surfaces).
\item[(ix)] 
Unlike the analogous situation of rationally connected varieties,
given a family of smooth, projective varieties in characteristic zero,
it is unclear whether the condition of being swept by rational
surfaces is a closed condition, or even an open condition, on fibers
of the family.
\end{enumerate}
\end{rmk}

\medskip\noindent 
It is technically more convenient to work with pencils of rational
curves than to work with rational surfaces.  Replacing $\PP^2$ by
the birational surface $\PP^1 \times \PP^1$, Definition~\ref{defn-manyRS}
can be rephrased 
in terms of pencils of rational curves.

\medskip\noindent
\begin{defn} \label{defn-sweeping} 
Let $X$ be a projective variety and $e\geq 1$ an integer.  An
integral, closed substack $Y \subset \Kbm{0,0}{X,e}$ is
\emph{sweeping} (resp. \emph{separably sweeping}) if,
\begin{enumerate}
\item[(i)] 
for a general geometric point of $Y$, the associated stable map
$f:C\rightarrow X$ has irreducible domain and is birational to its
image, and
\item[(ii)] 
the restriction over $Y$ of the universal morphism, $f:Y
\times_{\Kbm{0,0}{X,e}} \mc{C} \rightarrow X$ is surjective
(resp. surjective and separable).
\end{enumerate}
\end{defn}

\medskip\noindent
\begin{rmk} \label{rmk-sweeping} 
There are several obvious remarks.
\begin{enumerate}
\item[(i)]
This definition makes sense for any field $k$, not necessarily
algebraically closed nor
of characteristic $0$; although in case of positive characteristic one
should keep in mind that $\Kbm{0,0}{X,e}$ may only be an Artin
algebraic stack with finite diagonal (not a Deligne-Mumford stack).
\item[(ii)] 
If $Y \subset \Kbm{0,0}{X,e}$ is sweeping, then there is a Zariski
dense open substack that is a scheme.  In particular, it makes sense
to ask whether $Y$ is uniruled (resp. separably uniruled).
\item[(iii)] 
If $Y \subset \Kbm{0,0}{X,e}$ is sweeping (resp. separably sweeping),
then for the irreducible component $M \subset \Kbm{0,0}{X,e}$ 
containing $Y$, also $M$ is sweeping (resp. separably sweeping).
\item[(iv)]
If $X$ is uniruled (resp. separably uniruled), then there is an
integer $e$ and an irreducible component $M \subset \Kbm{0,0}{X,e}$
that is sweeping (resp. separably sweeping).
\end{enumerate}
\end{rmk}

\medskip\noindent  
\begin{lem} \label{lem-sweeping} 
Let $X$ be a projective variety over a field $k$ (not necessarily
algebraically closed nor of characteristic $0$).  The variety $X$ is
swept by rational surfaces (resp. separably swept by rational
surfaces) iff there exists an integer $e$ and a substack $Y \subset
\Kbm{0,0}{X,e}$ that is sweeping (resp. separably sweeping) such that
$Y$ is uniruled (resp. separably uniruled).
\end{lem}

\medskip\noindent
\begin{proof}
Let $X$ be a variety of dimension $n$ that is swept by rational
surfaces (resp. separably swept by rational surfaces).  Then there
exists a smooth quasi-projective variety $Z$ of dimension $n-2$ and a
rational transformation $F:Z \times \PP^1 \times \PP^1 \rightarrow X$
that is dominant (resp. dominant and separable) such that
$(\text{pr}_Z,F): Z \times \PP^1 \times \PP^1 \rightarrow Z \times X$
is generically finite (resp. birational to its image).  The
indeterminacy locus of $F$, call it $I \subset Z \times \PP^1 \times
\PP^1$, is a subvariety that has codimension at least $2$ at every
point.  The projection $\text{pr}_{12}(I) \subset Z \times \PP^1$ has
codimension at least $1$ at every point, i.e. it is contained in a
divisor $D \subset Z \times \PP^1$.  Denoting $U = Z\times \PP^1 - D$,
the rational transformation $F:U \times \PP^1 \rightarrow X$ is a
regular morphism.  After shrinking $U$ further, the morphism
$(\text{pr}_{12},F): U \times \PP^1 \rightarrow U \times X$ is a
finite morphism (resp. a finite, birational morphism).  Let $C'
\subset U \times X$ denote the image and let $G:C \rightarrow U \times
X$ denote the normalization of $C'$.  In case $X$ is separably swept
by rational surfaces, $G$ is the same as $(\text{pr}_{12},F)$.

\medskip\noindent 
After shrinking $U$ further, the morphism $\text{pr}_U:C \rightarrow
U$ is a smooth, proper morphism of relative dimension $1$.  Moreover
every geometric fiber is dominated by $\PP^1$, and therefore the
geometric fibers are connected curves that are isomorphic to $\PP^1$.
So $\text{pr}_U:C \rightarrow U$ is a family of genus $0$ curves and
$G:C \rightarrow U \times X$ is a family of stable maps of genus $0$
to $X$ of some degree $e$.  There is an induced $1$-morphism $\zeta:U
\rightarrow \Kbm{0,0}{X,e}$.  Define $Y \subset \Kbm{0,0}{X,e}$ to be
the closed image substack of $\zeta$.

\medskip\noindent 
The claim is that $Y$ is sweeping (resp. separably sweeping) and that
the coarse moduli space of $Y$ is uniruled (resp. separably uniruled).
There is an open substack of $Y$ over which the geometric fibers of
the universal curve are irreducible.  By construction, this open
substack contains the image of $\zeta$.  So it is Zariski dense in
$Y$.  Similarly, the condition that the stable maps are birational to
their images is an open condition on $Y$.  Since this condition holds
on the image of $\zeta$, it holds on an open dense subset of $Y$.
Therefore $Y$ satisfies Item (i) of Definition~\ref{defn-sweeping}.
Now $F: U \times \PP^1 \rightarrow X$ is dominant. Therefore also $G:C
\rightarrow X$ is dominant.  So $f:Y \times_{\Kbm{0,0}{X,e}} \mc{C}
\rightarrow X$ is dominant.  Therefore $Y$ satisfies Item (ii) of
Definition~\ref{defn-sweeping}, i.e. $Y$ is sweeping.  Moreover if $X$
is separably swept by rational surfaces, then $F$ is dominant,
generically finite and separable which implies that also $f:Y
\times_{\Kbm{0,0}{X,e}} \mc{C} \rightarrow X$ is dominant, generically
finite and separable (since a sub-extension of a separable field
extension is separable), so $Y$ is separably sweeping.

\medskip\noindent
Consider $\zeta:U \rightarrow Y$.  The claim is that $\zeta$ is
dominant and generically finite (resp. dominant, generically finite
and separable).  There is a factorization of $F$ of the form
\begin{equation}
U\times \PP^1 \rightarrow U \times_Y
(Y \times_{\Kbm{0,0}{X,e}} \mc{C}) = C \rightarrow Y
\times_{\Kbm{0,0}{X,e}} \mc{C} \xrightarrow{f} X.
\end{equation}
Since $F$ is dominant and generically finite (resp. dominant,
generically finite and separable), each of these factors of $F$ is
dominant and generically finite (resp. dominant, generically finite
and separable).  In particular $U \times_Y (Y \times_{\Kbm{0,0}{X,e}}
\mc{C}) \rightarrow Y \times_{\Kbm{0,0}{X,e}} \mc{C}$ is dominant and
generically finite (resp. and separable).  But this is just the
base-change of $\zeta:U \rightarrow Y$ by the smooth surjective
morphism $\text{pr}_Y: Y \times_{\Kbm{0,0}{X,e}} \mc{C} \rightarrow
Y$.  Therefore $\zeta:U \rightarrow Y$ is dominant and generically
finite (resp. dominant, generically finite and separable).  Since $U$
is an open subset of $Z \times \PP^1$, $Y$ is
uniruled (resp. separably uniruled).  This proves the forward
direction of the lemma.

\medskip\noindent
Conversely, suppose that $Y \subset \Kbm{0,0}{X,e}$ is sweeping
(resp. separably sweeping) and $Y$ is uniruled (resp. separably
uniruled).  Then there exists a rational transformation $\zeta:Z \times
\PP^1 \rightarrow Y$ that is dominant and generically finite
(resp. dominant, generically finite and separable).

\medskip\noindent
Let $\pi:C \rightarrow Z \times \PP^1$ be a projective completion of
the pullback of the universal family of curves over $\Kbm{0,0}{X,e}$,
and let $H:C \rightarrow X$ be the pullback of the universal stable
map.  After blowing up $C$, $C$ can be made normal
and $H$ can be made a regular morphism.  Moreover, over a dense open
subset of $Z\times \PP^1$, the projection $\pi$ is a smooth, proper
morphism whose geometric fibers are connected curves of genus $0$.  So
the geometric generic fiber of $\text{pr}_Z\circ \pi:C \rightarrow Z$
is a conic bundle over $\PP^1$.  By Tsen's theorem
(c.f. ~\cite[Cor. IV.6.6.2]{K}), the base-change $C \otimes_{K(Z)}
\overline{K}(Z)$ is a rational surface.  By ~\cite{Coombes}, in fact
there is a dominant, generically finite and separable morphism
$Z'\rightarrow Z$ and a birational transformation over $Z'$, $G:Z'
\times \PP^2 \rightarrow Z' \times_Z C$.  Observe that the composition
$Z' \times \PP^1 \rightarrow Z'\times \PP^1 \rightarrow Y$ is still
dominant and generically finite (resp. dominant, generically finite
and separable).  Therefore $Z$ may be replaced by $Z'$ so that $C$ is
birational to $Z \times \PP^2$ over $Z$.

\medskip\noindent
Denote by $F$ the rational transformation $H\circ G: Z \times \PP^2
\rightarrow X$.  There is a factorization of $F$ of the form
\begin{equation}
Z \times \PP^2 \xrightarrow{G} C \rightarrow Y \times_{\Kbm{0,0}{X,e}}
\mc{C} \xrightarrow{f} X.
\end{equation}
By assumption, each of these factors is dominant (resp. dominant and
separable).  Therefore $F$ is dominant (resp. dominant and separable).
Consider $(\text{pr}_Z \circ \pi,F)\circ G:Z \times \PP^2 \rightarrow
Z\times X$.  By the hypotheses on $Y$, this morphism is generically
finite (resp. generically finite and separable to its image).
Therefore $X$ is swept by rational surfaces (resp. separably swept by
rational surfaces).  This finishes the proof of the lemma.
\end{proof}

\medskip\noindent
Because of the lemma, it is natural to try to understand the sweeping
substacks of $\Kbm{0,0}{X,e}$, and in particular to try to understand
the Kodaira dimension of these substacks.  Recall that a standard
conjecture from the minimal model program predicts that an algebraic
variety is uniruled iff its Kodaira dimension is negative.  If
$X\subset \PP^n$ is a hypersurface of degree $d \leq n$, then for $e
>> 0$, an irreducible component of $\Kbm{0,0}{X,e}$ will itself be
sweeping.  So the first step is to determine the Kodaira dimension of
each of the irreducible components of $\Kbm{0,0}{X,e}$ that is sweeping.

\medskip\noindent
As mentioned above, for $X$ a general hypersurface of degree $d <
\frac{n+1}{2}$ the stacks $\Kbm{0,0}{X,e}$ are irreducible and
reduced, and the same is conjectured whenever $d < n-1$.  Under the
hypothesis that each $\Kbm{0,0}{X,e}$ is an integral, normal stack of
the expected dimension, one can compute the canonical divisor of
$\Kbm{0,0}{X,e}$, i.e. one can compute the \emph{expected canonical
divisor}.  This is carried out in Section~\ref{sec-canKBM}, and is a
straightforward extension of the derivation in ~\cite{Pand97}.  The
exciting observation is that when $d$ satisfies the inequalities
$d\leq n-4$ and $d^2 \geq n+2$, then for almost all values of $e$, the
canonical divisor is \emph{big}.  And when $d\leq n-7$ and $d(d+1)
\geq 2(n+1)$, then the canonical divisor is big for every $e\geq 1$.

\medskip\noindent
Recall that a sufficient condition for a variety $M$ to be of general
type is that the canonical divisor $K_M$ is big and $M$ has only
canonical singularities.  This raises the hope that in the degree
range above the spaces $\kbm{0,0}{X,e}$ are all of general type.  The
missing ingredient is an analysis of the singularities of
$\kbm{0,0}{X,e}$.  This paper is the result of an ``initial
investigation'' of the singularities of $\kbm{0,0}{X,e}$.  Obviously
much work is still needed to prove that $\kbm{0,0}{X,e}$ contains no
uniruled sweeping subvariety.

\subsection{Detailed summary} \label{subsec-summary} 

\medskip\noindent
The proof of Theorem~\ref{thm-main} is a
deformation-and-specialization argument. The stack
$\Kbm{0,0}{\PP^n,e}$ is smooth, therefore the singularities of $C_d$
come from loci in $\Kbm{0,0}{\PP^n,e}$ over which the fiber dimension
of $\pi_d:C_d \rightarrow \Kbm{0,0}{\PP^n,e}$ jumps.  This defines a
stratification of $\Kbm{0,0}{\PP^n,e}$, and the ``deepest'' stratum
corresponds to the locus $Y \subset \Kbm{0,0}{\PP^n,e}$ parametrizing
multiple covers of lines.  This stratum is a smooth variety, and the
normal bundle of $Y \subset \Kbm{0,0}{\PP^n,e}$ is essentially the bundle 
of all
$(e-1) \times (n-1)$ matrices.

\medskip\noindent
The normal cone of $\pi_d^{-1}(Y) \subset C_d$ is a projective cone
over the normal bundle of $Y \subset \Kbm{0,0}{\PP^n,e}$.  When $d+e
\leq n$ this projective cone is even a \emph{projective Abelian cone}
associated to a torsion-free sheaf on the normal bundle of $Y \subset
\Kbm{0,0}{\PP^n,e}$.  This torsion-free sheaf is essentially the
direct sum of $d$ copies of the quotient of a bundle of rank $(n-1)$
by the universal $(e-1) \times (n-1)$ matrix mentioned above.  By an
explicit resolution of singularities, this
projective Abelian cone is canonical. Then \emph{deformation to the
normal cone} produces a family over $\PP^1$ whose fibers over $\AAA^1$
are all isomorphic to $C_d$ and whose fiber over $\infty$ is the
normal cone of $\pi_d^{-1}(Y)$.

\medskip\noindent
Applying \emph{inversion-of-adjunction} results to this family, 
there exists an open substack $U \subset
\Kbm{0,0}{\PP^n,e}$ containing $Y$ such that $\pi_d^{-1}(U)$ is
canonical.  The action of the group $\text{GL}_{n+1}$ on
$\PP^n$ induces an action of $\text{GL}_{n+1}$ on
$\Kbm{0,0}{\PP^n,e}$.  The open substack $U$ is
$\text{GL}_{n+1}$-invariant, but also $Y$ intersects the closure of
\emph{every} orbit of $\text{GL}_{n+1}$ on $\Kbm{0,0}{\PP^n,e}$.  Therefore
$U$ is all of $\Kbm{0,0}{\PP^n,e}$, proving that $C_d$ is
canonical.

\medskip\noindent
Section ~\ref{sec-discrep}, Section ~\ref{sec-deter}, Section
~\ref{sec-spec} and Section ~\ref{sec-dnc} are all of a foundational
nature, proving basic results about the singularities of the relative
Grassmannian, or \emph{Grassmannian cone}, associated to a
torsion-free coherent sheaf $\mc{E}$ which has (local) projective
dimension $1$.  The main result of Section~\ref{sec-discrep} is
Proposition~\ref{prop-relcan}: This relates the singularities of a
Grassmannian cone $C$ parametrizing rank $r$ locally free quotients of
$\mc{E}$ to the singularities of the pair $(B,r\cdot B_{g-1})$, where
$B_{g-1}$ is the closed subscheme determined by the \emph{Fitting
ideal} of $\mc{E}$.  The main result of Section~\ref{sec-deter} is
Proposition~\ref{prop-cone}: This computes the singularities of the
Grassmannian cone of a direct sum of $a$ copies of the cokernel of the
universal $g\times f$ matrix on the affine space of $g \times f$
matrices.  The main result of Section~\ref{sec-spec} is
Corollary~\ref{cor-spec}: This applies known results related to
inversion-of-adjunction to prove adjunction results for a pair
$(B, r\cdot B_{g-1})$ as above.  Section~\ref{sec-dnc} is a review of
the construction of deformation to the normal cone in preparation for
the proof of the main theorem.

\medskip\noindent
In Section~\ref{sec-covers} and Section~\ref{sec-proof} the proof of
the main theorem is given.  Section~\ref{sec-covers} introduces the
closed substack $Y \subset \Kbm{0,0}{\PP^n,e}$ parametrizing multiple
covers of lines.  The main result is Proposition~\ref{prop-phid}, which
is an analysis of the coherent sheaves used to define $C_d$ when
restricted to a first-order neighborhood of $Y$ in
$\Kbm{0,0}{\PP^n,e}$.  Section~\ref{sec-proof} gives the proof of
Theorem~\ref{thm-main} along the lines discussed above.

\medskip\noindent
In Section~\ref{sec-RSBT}, the Reid--Shepherd-Barron--Tai
criterion is used to prove that the coarse moduli space $\kbm{0,0}{\PP^n,e}$
has only canonical singularities (and in most cases it is even
terminal).  Combining this analysis with the proof of
Theorem~\ref{thm-main}, Section~\ref{sec-conj} proves that the
inversion-of-adjunction conjecture of Koll\'ar and Shokurov implies
that the coarse moduli space of the stack $C_d$ has only canonical
singularities when either $e\geq 3$ and $d+e \leq n$ or $e=2$ and $d+3
\leq n$.

\medskip\noindent
Finally, in Section~\ref{sec-canKBM}, the expected
canonical divisor of the stack $\Kbm{0,0}{X,e}$ is computed -- 
this is the same as
the canonical divisor on the coarse moduli space $\kbm{0,0}{X,e}$ in
most cases.  When $n +1 < d^2 < (n-3)^2$, for most
choices of $e$ the expected canonical divisor of $\kbm{0,0}{X,e}$ is
big.

\medskip\noindent
\textbf{Acknowledgments}  
This paper is a continuation of ~\cite{HRS2} and ~\cite{HS2}.  My
greatest debt is to my coauthors Joe Harris and Mike Roth.  I am also
grateful for useful conversations with Jiun-Cheng Chen, A. Johan de
Jong, Mircea Musta\c{t}\v{a}, and especially J\'{a}nos Koll\'{a}r.  I
was supported by NSF Grant DMS-0201423.


\section{Discrepancies of a Grassmannian cone} \label{sec-discrep}

\medskip\noindent
Let $B$ be a Noetherian scheme that is connected, normal, and
$\QQ$-Gorenstein of pure dimension $b$.  Let $\phi:\mc{G} \rightarrow
\mc{F}$ be a morphism of locally free $\OO_B$-modules of rank $g$ and
$f$ respectively such that the cokernel $\mc{E} = \text{Coker}(\phi)$
has generic rank $e=f-g$.  In this section all results are of a local
nature on $B$.  So the results apply equally well to a coherent
sheaf $\mc{E}$ that has local projective dimension $1$ (in the sense
of ~\cite[p. 280]{Ma}).

\medskip\noindent
\begin{notat} \label{notat-C} 
Denote by $\text{det}(\mc{E})$ the invertible sheaf
$\text{det}(\mc{F}) \otimes_{\OO_B} \text{det}(\mc{G})^\vee$.  Let $r$
be an integer, $1\leq r \leq e$, and denote by the pair $(\pi:C
\rightarrow B, \alpha:\pi^* \mc{E} \rightarrow \mc{Q})$ the relative
Grassmannian cone over $B$ parametrizing rank $r$ locally free
quotients of $\mc{E}$.  Denote by $\OO_C(1)$ the invertible sheaf on
$C$, $\text{det}(\mc{Q})$.  Denote by the pair $(\rho:C' \rightarrow
B, \beta:\rho^* \mc{F} \rightarrow \mc{Q}')$ the relative Grassmannian
bundle over $B$ parametrizing rank $r$ locally free quotients of
$\mc{F}$.  Denote by $\OO_{C'}(1)$ the invertible sheaf on $C'$,
$\text{det}(\mc{Q}')$.  The surjection $\pi^* \mc{F} \rightarrow \pi^*
\mc{E} \xrightarrow{\alpha} \mc{Q}$ induces a morphism of $B$-schemes
which we denote $h:C \rightarrow C'$.
\end{notat}

\medskip\noindent
Comparing universal properties, $h$ is a closed
immersion whose ideal sheaf is the image of the composite morphism
\begin{equation}
\rho^* \mc{G} \otimes (Q')^\vee \xrightarrow{\phi
  \otimes \beta^\dagger}  \pi^*\mc{F} \otimes_{\OO_{C'}} \rho^*\mc{F}^\vee
  \xrightarrow{\text{Trace}} \OO_{C'}.
\end{equation}
This morphism is denoted by $\gamma$.

\medskip\noindent
If $\mc{E}$ is locally free, then $\pi:C \rightarrow B$
is Zariski locally a Grassmannian bundle.  But, in general, the fiber
dimension is not necessarily constant.  
The results of the next few sections hold for arbitrary integer $r$; although
the case of interest in the
remainder of the paper is $r=1$.  For $r=1$, $\pi:C \rightarrow B$
is a \emph{projective Abelian cone} (paraphrasing notation of
\cite{F} and \cite{BF}).  
We begin with an obvious criterion for $C$ to be
irreducible.

\medskip\noindent
\begin{notat} \label{notat-Bk} 
Let $k$ be an integer $k=0,\dots,g$.  Denote by $B_k \subset B$ the
closed subscheme whose ideal sheaf is generated by the $(k+1)\times
(k+1)$-minors of $\phi$, i.e. $B_k$ is the locus where $\phi$ has rank
at most $k$.
\end{notat}  

\medskip\noindent
\begin{lem} \label{lem-codims} 
\begin{enumerate}
\item[(i)]
The coherent sheaf $\mc{E}$ is torsion-free iff
$\text{codim}_B(B_{g-1}) \geq 2$.  
\item[(ii)]
The scheme $C$ is irreducible iff
$\text{codim}_B(B_k - B_{k-1}) \geq r(g-k) +1$ for all
$k=0,\dots,g-1$, in which case $C$ has dimension $c=b+r(e-r)$.
\item[(iii)]
Furthermore, the scheme $C$ is regular in codimension $1$ points if
$\text{codim}_B(B_k - B_{k-1}) \geq r(g-k)+2$ for all $k=0,\dots,g-1$.
\end{enumerate}
If $g=0$, all of these conditions are vacuously satisfied.
\end{lem}

\begin{proof}
\textbf{(i)}:
Torsion sections of the sheaf $\mc{E}$ correspond locally on $B$ to
sections of $\mc{F}$ that are generically in the image of $\mc{G}$.
Since $B$ is normal and since $\mc{G}$ is locally free, the image of
$\mc{G}$ in $\mc{F}$ equals the intersection of its localization at
all codimension $1$ points of $S$.  Hence a section of $\mc{F}$
corresponds to a torsion (resp. torsion-free) section of 
$\mc{E}$ iff its localization
at all codimension $1$ points of $B$ is torsion (resp. torsion-free) 
in $\mc{E}$.
Therefore $\mc{E}$ is torsion-free iff $\text{codim}_B(B_{g-1}) \geq
2$.

\medskip\noindent
\textbf{(ii)}:
Of course $C'$ has pure dimension $b+r(f-r)$ at every point.  Consider
the morphism $\gamma$.  Since the rank of $\rho^* \mc{G} \otimes
(\mc{Q}')^\vee$ is $g\cdot r$, the dimension of $C$ at every point is
at least $c:= b + r(f-r) - g\cdot r = b + r(e-r)$.

\medskip\noindent
The restriction of $\pi$ over the locally closed subscheme
$B_k-B_{k-1}$ is proper and smooth of relative dimension $r\lt( (f-k)
- r \rt)$ and has geometrically irreducible (and nonempty) fibers.  In
particular, the preimage of $B-B_{g-1}$ is normal and irreducible of
dimension $c= b+r(e-r)$.  Therefore, to prove that $C$ is
irreducible, it suffices to prove that for each $k=0,\dots, g-1$, the
dimension of $\pi^{-1}(B_k - B_{k-1})$ is at most $c-1$.  Conversely,
if any of these sets has dimension $c$ or greater, then it is not in
the closure of $\pi^{-1}(B-B_{g-1})$, and therefore $C$ is
reducible.  Hence $C$ is irreducible iff $\text{dim} \pi^{-1}(B_k
- B_{k-1}) \leq c-1$.  The dimension of this set is clearly $c -
\lt[ \text{codim}_B \lt(B_k - B_{k-1} \rt) -r(g-k) \rt]$.  Therefore 
$C$ is
irreducible iff $\text{codim}_B \lt(B_k - B_{k-1} \rt) \geq r(g-k) +
1$ for all $k=0,\dots, g-1$.

\medskip\noindent
\textbf{(iii)}:
Now suppose that in fact $\text{codim}_B \lt(B_k - B_{k-1} \rt) \geq
r(g-k) + 2$ for all $k=0,\dots,g-1$.  The preimage $\pi^{-1}(B -
B_{g-1})$ is normal.  So to prove that $C$ is regular in codimension
$1$ points, it suffices to prove that $C$ is regular in codimension
$1$ points that are contained in one of the subsets $\pi^{-1}(B_k -
B_{k-1})$ for $k=0,\dots, g-1$.  The inequality guarantees that
each of the sets $\pi^{-1}(B_k - B_{k-1})$ has codimension at least
$2$ in $C$, therefore there are no such codimension $1$ points.
\end{proof}

\begin{rmk} \label{rmk-codims} 
It can happen that $\phi$ satisfies the inequality of (ii) so that $C$
is irreducible, and yet $C$ is not regular in codimension $1$. For
example, let $B= \AAA^2$, let $r=1$, and let $\phi$ be
the morphism $\phi:\OO_{\AAA^2} \rightarrow
\OO_{\AAA^2}^{\oplus 2}$ with matrix $(x^2,y^2)^\dagger$.  On the
other hand, the third inequality is not a necessary condition,
cf. Proposition~\ref{prop-cone}.
\end{rmk}

\medskip\noindent
\begin{hyp} \label{hyp-tf} 
Unless stated otherwise, the coherent sheaf $\mc{E}$ is
torsion-free, i.e. $\text{codim}_B(B_{g-1}) \geq 2$.
\end{hyp}

\medskip\noindent
\begin{lem} \label{lem-discrep} 
Suppose that $C$ has pure dimension $c=b+r(e-r)$.
\begin{enumerate}
\item[(i)] 
If $B$ is Cohen-Macaulay, then $C$ is Cohen-Macaulay.  If also $C$ is
regular in codimension $1$, then $C$ is normal.
\item[(ii)] 
If $B$ is Gorenstein, then $C$ is Gorenstein.
\item[(iii)] 
The morphism $\pi$ admits a relative dualizing complex of the form
$\omega_\pi[r(e-r)]$ where $\omega_\pi$ is
the invertible sheaf $\pi^*\text{det}(\mc{E})^{\otimes r}
\otimes_{\OO_C} \OO_C(-e)$.  
\item[(iv)] 
If $C$ is normal, then $C$ is $\QQ$-Gorenstein and the $\QQ$-Cartier
divisor class $K_C$ equals $\pi^*K_B + K_\pi$ where $K_\pi$ is the
divisor class of $\omega_\pi$.
\end{enumerate}
\end{lem}

\medskip\noindent
\begin{proof}
\textbf{(i)}:
By assumption $h(C)$ has pure codimension $g\cdot r$ in $C$, which
equals the rank of $\mc{G}\otimes_{\OO_{C'}} (Q')^\vee$.  By
~\cite[Thm. 17.3 and Thm. 17.4]{Ma}, if $B$ is Cohen-Macaulay then
$h(C)$ is Cohen-Macaulay.  It follows from Serre's
criterion, ~\cite[Thm. 23.8]{Ma}, that if $C$ is regular in codimension
$1$, then $C$ is normal.  

\medskip\noindent
\textbf{(ii)}:
Using ~\cite[Exer. 18.1]{Ma}, if $B$ is
Gorenstein then $h(C)$ is Gorenstein.

\medskip\noindent
\textbf{(iii)}:
The morphism $\rho$ is smooth and has a relative dualizing complex
$\omega_\rho[r(f-r)]$ where $\omega_\rho$ is the invertible sheaf
$\rho^*\text{det}(\mc{F})^{\otimes r} \otimes_{\OO_{C'}}
\OO_{C'}(-f)$.  The morphism $h$ is a regular embedding and has a
relative dualizing complex $\omega_h[-rg]$ where $\omega_h$ is the
pullback of the invertible sheaf $\textit{Ext}^{rg}_{\OO_{C'}}
(h_*\OO_C, \OO_{C'})$.  Forming the Koszul complex associated to the
sheaf map $\gamma$, $\omega_h$ is isomorphic to
$\pi^*\text{det}\lt( \mc{G}^\vee \rt)^{\otimes r} \otimes_{\OO_C}
\OO_{C}(g)$.  Therefore the composite $\pi = \rho\circ h$ has a
relative dualizing complex $\omega_\pi[r(e-r)]$ where $\omega_\pi$ is
an invertible sheaf isomorphic to $\pi^*\text{det}(\mc{E})^{\otimes r}
\otimes \OO_C(-e)$.

\medskip\noindent
\textbf{(iv)}:
Suppose that $C$ is normal.  Let $U \subset C$ be the smooth locus of
$C$ and let $V\subset C'$ denote the smooth locus of $C'$.  Since $h$
is a regular embedding, $U \subset h^{-1}(V)$.  Also,
since $\rho$ is smooth, $V$ is just $\rho^{-1}(W)$, where
$W\subset B$ is the smooth locus.  Now $\omega_B|_W$ is isomorphic to
$\OO_B(K_B)|_W$ and $\omega_{C'}|_{\rho^{-1}(W)}$ is isomorphic to
$\rho^*\OO_B(K_B) \otimes_{\OO_{C'}} \omega_\rho$.  By the same
reasoning as above, $\omega_C|_V \cong
h^*\rho^*\OO_B(K_B) \otimes \omega_\pi$.  But of course $\omega_C|_V$
is isomorphic to $\OO_C(K_C)|_V$.  Therefore the $\QQ$-Weil divisor
class $K_C$ is equal to $\rho^*K_C + K_\pi$ where $K_\pi$ is the
divisor class of $\omega_\pi$.  Since this is a $\QQ$-Cartier divisor
class, $C$ is $\QQ$-Gorenstein.
\end{proof}

\medskip\noindent
\begin{cor} \label{cor-discrep} 
Let $Y \subset B$ be a closed subscheme that is a regular embedding
of pure codimension $\text{codim}_B(Y)$, i.e. for every closed point
$p\in Y$, the ideal sheaf $\mc{I}_p \subset \OO_{B,p}$ is generated by
a regular sequence of length $\text{codim}_B(Y)$.
\begin{enumerate}
\item[(i)]
If $C \times_B Y$ has the expected dimension $b-\text{codim}_B(Y) +
r(e-r)$, then there exists an open subset $U\subset B$ containing $Y$
such that $C \times_B U$ has the expected dimension $b + r(e-r)$.
\item[(ii)]
If also $C\times_B Y$ is irreducible, then $U$ can be chosen so that
$C \times_B U$ is irreducible.
\item[(iii)]
If also $C\times_B Y$ is normal and $B$ is Cohen-Macaulay, then 
$U$ can be chosen so that $C\times_B U$ is normal.
\end{enumerate}
\end{cor}

\medskip\noindent
\begin{proof}
\textbf{(i)}:
Let $C_i \subset C$ be an irreducible component of $C$ that has
nonempty intersection with $C\times_B Y$.  The dimension of $C$ is 
\emph{at
least} $b+r(e-r)$; the claim is that it is exactly
$b+r(e-r)$.  Since $Y \subset B$ is a regular embedding locally
defined by a regular sequence of length $\text{codim}_B(Y)$, it
follows by Krull's Hauptidealsatz that $\text{dim}(C_i \times_B Y)
\geq \text{dim}(C_i) - \text{codim}_B(Y)$.  On the other hand, since
$C_i \times_B Y$ is a closed subscheme of $C \times_B Y$,
$\text{dim}(C_i \times_B Y) \leq \text{dim}(C \times_B Y) = b+r(e-r) -
\text{codim}_B(Y)$.  Therefore $\text{dim}(C_i) = b +
r(e-r)$.  So for any irreducible component $C_i \subset C$ whose
dimension is larger than $b+r(e-r)$, $\pi(C_i) \cap Y =
\emptyset$.  Define $U$ to be the complement of the finitely many
closed sets $\pi(C_i)$ as above.  Then $U$ satisfies (i).

\medskip\noindent
\textbf{(ii)}:
Suppose that also $C \times_B Y$ is irreducible.  By
Lemma~\ref{lem-codims}, for each $k=0,\dots,g-1$, 
$\text{codim}_Y(Y_k - Y_{k-1}) \geq r(g-k) + 1$.  Now $Y_k = B_k \cap
Y$.  So again by Krull's Hauptidealsatz, for every irreducible
component $(B_k)_i$ of $B_k$ that intersects $Y$,
$\text{dim}((B_k)_i) \leq \text{dim}(Y_k) + \text{codim}_B(Y)$,
i.e. $\text{codim}_B((B_k)_i) \geq \text{codim}_Y(Y_k)$.  Now
shrink the $U$ from the last paragraph by taking the complement of the
finitely many irreducible components $(B_k)_i$ that do not intersect
$Y$ and have the wrong codimension.  Then for every $(B_k)_i$
that intersects $U$, $\text{codim}_B((B_k)_i) \geq
\text{codim}_Y(Y_k) \geq r(g-k) + 1$.  Therefore, by
Lemma~\ref{lem-codims}, $C \times_B U$ is irreducible.

\medskip\noindent
\textbf{(iii)}:
Finally, suppose that also $C\times_B Y$ is normal and $B$ is
Cohen-Macaulay.  By Item $(i)$ of Lemma~\ref{lem-discrep}, to prove
that $C \times_B U$ is normal, it suffices to prove that $C \times_B U$
is regular in codimension one.  Let $(C\times_B U)_{\text{sing}}$ be
the singular locus of $C\times_B U$.  Since $C\times_B Y \subset
C\times_B U$ is a Cartier divisor, every regular point of $C\times_B
Y$ is also a regular point of $C\times_B U$.  Therefore the
intersection of $(C\times_B U)_{\text{sing}}$ with $C\times_B Y$ is
contained in $(C\times_B Y)_{\text{sing}}$.  Since $C\times_B Y$ is
normal, $(C\times_B Y)_{\text{sing}}$ has codimension at least $2$ in
$C\times_B Y$.  So, again by Krull's Hauptidealsatz, every
irreducible component of $(C\times_B U)_{\text{sing}}$ that
intersects $C\times_B Y$ has codimension at least $2$ in $C\times_B
U$.  After shrinking $U$ more, $C\times_B U$ is
normal.
\end{proof}

\medskip\noindent
\begin{hyp} \label{hyp-1} 
Unless stated otherwise, $C$ is irreducible of the expected
dimension $c = b + r(e-r)$.
\end{hyp}

\medskip\noindent
\begin{defn} \label{defn-res} 
A morphism of schemes $u:\widetilde{B} \rightarrow B$ is a
\emph{resolution of} $\mc{E}$ if,
\begin{enumerate}
\item[(i)] 
$u$ is a birational, proper morphism,
\item[(ii)] 
$\widetilde{B}$ is smooth, 
\item[(iii)] 
the exceptional locus of $u$ is a simple normal crossings
  divisor $E_1 \cup \dots \cup E_k$, and
\item[(iv)] 
the coherent sheaf $\widetilde{\mc{E}} :=
  u^*\mc{E}/\text{torsion}$ is a locally free
  $\OO_{\widetilde{B}}$-module of rank $e$.
\end{enumerate}
\end{defn}

\medskip\noindent
\begin{notat} \label{notat-res} 
Let $u:\widetilde{B} \rightarrow B$ be a resolution of $\mc{E}$.
Denote by $\widetilde{\mc{G}}$ the kernel of the induced surjective
sheaf map $u^*\mc{F} \rightarrow \widetilde{\mc{E}}$ and denote by
$\widetilde{\phi}: \widetilde{\mc{G}} \rightarrow u^* \mc{F}$ the
induced injection, i.e. $\widetilde{\mc{E}}$ is the cokernel of
$\widetilde{\phi}$.  Denote by the pair $(\text{pr}_1:\widetilde{B}
\times_B C' \rightarrow \widetilde{B}, \text{pr}_2^*\beta:
\text{pr}_2^*\rho^*\mc{F} \rightarrow \text{pr}_2^*\mc{Q}')$ the
base-change of $(\rho:C' \rightarrow B, \beta)$.  Denote by the pair
$(\widetilde{\pi}:\widetilde{C} \rightarrow \widetilde{B},
\widetilde{\alpha}:\widetilde{\pi}^* \widetilde{\mc{E}} \rightarrow
\widetilde{Q})$ the Grassmannian bundle parametrizing rank $r$ locally
free quotients of $\widetilde{\mc{E}}$.  Denote by
$\OO_{\widetilde{C}}(1)$ the invertible sheaf on $\widetilde{C}$,
$\text{det}(\widetilde{Q})$.  The surjection $u^*\mc{F} \rightarrow
\widetilde{\mc{E}}$ induces a closed immersion that is denoted
$\widetilde{h}:\widetilde{C} \rightarrow \widetilde{B} \times_B C'$.
\end{notat}

\medskip\noindent
Because the morphism $u^*\mc{F} \rightarrow \widetilde{\mc{E}}$
factors through the pullback $u^*\mc{F} \rightarrow u^*\mc{E}$, the
composition $\text{pr}_2\circ h: \widetilde{C} \rightarrow C'$ factors
through $h$, i.e. there is an induced morphism $v:\widetilde{C}
\rightarrow C$.  Of course $v^* \mc{Q} \cong \widetilde{\mc{Q}}$ and
$\pi\circ v = u\circ \widetilde{\pi}$.

\medskip\noindent
\begin{lem} \label{lem-res} 
The morphism $v: \widetilde{C} \rightarrow C$ is a weak resolution of
singularities, that is
\begin{enumerate}
\item[(i)]
$v$ is a proper, birational morphism, and
\item[(ii)] 
$\widetilde{C}$ is nonsingular.
\end{enumerate} 
Moreover, the exceptional locus of $v$ is contained in the divisor
$\widetilde{\pi}^{-1}(E_1 \cup \dots \cup E_k)$.
\end{lem}

\medskip\noindent
\begin{proof}
This is obvious.
\end{proof}

\medskip\noindent
\begin{rmk} \label{rmk-discrep} 
If $B$ is a finite type scheme over an algebraically closed field of
characteristic $0$, then a resolution of $\mc{E}$ exists.  Form the
Grassmannian bundle $(\pi_e:G\rightarrow B,\alpha_e: \pi_e^* \mc{E}
\rightarrow Q)$ parametrizing rank $e$ locally free quotients of
$\mc{E}$.  Since $\mc{E}$ generically has rank $e$, there is an
irreducible component $G_0$ of $G$ such that $\pi_e:G_0 \rightarrow B$
is birational.  Let $Z\subset G_0$ denote the fundamental locus of
$\pi_e^{-1}$.  By ~\cite{Hir}, there exists a log resolution
$\widetilde{B} \rightarrow B$ of the pair $(G_0,Z)$, and the induced
morphism $u:\widetilde{B} \rightarrow B$ will be a resolution of $\mc{E}$.
\end{rmk}

\medskip\noindent
\begin{lem} \label{lem-logres} 
The morphism $u:\widetilde{B} \rightarrow B$ is a log resolution of
the pair $(B,B_{g-1})$.
\end{lem}

\medskip\noindent
\begin{proof}
The morphism $\phi:\mc{G}\rightarrow \mc{F}$ induces an element
$\bigwedge^g\phi \in \textit{Hom}_{\OO_B}(\bigwedge^g
\mc{G},\bigwedge^g \mc{F})$, i.e. an element in $\bigwedge^g \mc{F}
\otimes_{\OO_B} \lt(\bigwedge^g \mc{G} \rt)^\vee$.  There is an
induced map
\begin{equation}
\text{Id}\otimes \bigwedge^g \phi: \bigwedge^e \mc{F} \rightarrow
\bigwedge^e \mc{F} \otimes_{\OO_B} \bigwedge^g \mc{F} \otimes_{\OO_B}
\lt( \bigwedge^g \mc{G} \rt)^\vee.
\end{equation}
Compose this map with the wedge product map on $\mc{F}$ to get
a map $\bigwedge^e \mc{F} \rightarrow \text{det}(\mc{E})$.  Consider
the restriction of this map to the torsion-free subsheaf $\mc{G}\cdot
\bigwedge^{e-1} \mc{F} \subset \bigwedge^e \mc{F}$.  On the generic
point of $B$, it is clear that this map is the zero map.  A
morphism between torsion-free sheaves on $B$ that is zero at the
generic point of $B$ is the zero map.  So the restriction of the map
to $\mc{G} \cdot \bigwedge^{e-1} \mc{F}$ is zero, proving that
the map factors through a map $\psi: \bigwedge^e \mc{E} \rightarrow
\text{det}(\mc{E})$.

\medskip\noindent 
Since $\mc{E}$ is torsion-free, $B_{g-1}$ has codimension at least $2$
in $B$.  So $\psi$ is an isomorphism in codimension $1$.  Denote by
$\mc{I} \subset \OO_B$ the unique ideal sheaf so that
$\text{Image}(\psi)$ equals $\mc{I} \cdot \text{det}(\mc{E})$.  In
other words, $\mc{I}$ is the $e^{\text{th}}$ \emph{Fitting ideal} of
$\phi$ -- the ideal sheaf generated by the $g\times g$ minors of
$\phi$.  This is precisely the ideal sheaf of the subscheme $B_{g-1}$.
It follows that $\lt( u^*\bigwedge^e \mc{E} \rt)/\text{torsion}$ is
just $u^{-1}\mc{I}\cdot u^*\text{det}(\mc{E})$.

\medskip\noindent
The surjection $u^*\mc{E} \rightarrow \widetilde{\mc{E}}$ induces a
surjection $\lt(u^*\bigwedge^e \mc{E}\rt)/\text{torsion} \rightarrow
\bigwedge^e \widetilde{\mc{E}}$.  It is easy to see that this
surjection is in fact an isomorphism.  Hence there is a canonical
isomorphism $u^{-1}\mc{I}\cdot u^*\text{det}(\mc{E}) \cong \text{det}
(\widetilde{\mc{E}})$.  In particular, the pullback ideal sheaf
$u^{-1}\mc{I}\cdot \OO_{\widetilde{B}}$ is a Cartier divisor.
Moreover, this divisor is a subdivisor of the simple normal crossings
divisor $E_1 \cup \dots \cup E_k$, and so it is also a simple normal
crossings divisor.  
\end{proof}

\medskip\noindent
By definition, the log discrepancies $a(E_i;B,r\cdot B_{g-1})$ of
$(B,r\cdot B_{g-1})$ along the divisors $E_1,\dots,E_k \subset
\widetilde{B}$ are defined by
\begin{equation}
K_{\widetilde{B}} - u^*K_B - r\cdot u^{-1}(B_{g-1}) = \sum_{i=1}^k \lt(
a(E_i;B,r\cdot B_{g-1})
-1 \rt) E_i
\end{equation}
where $u^{-1}(B_{g-1}) \subset \widetilde{B}$ is defined to be the
closed subscheme corresponding to $u^{-1}\mc{I} \cdot
\OO_{\widetilde{B}}$.  

\medskip\noindent
\begin{lem}  \label{lem-relcan} 
The relative canonical divisor of $v:\widetilde{C} \rightarrow C$ is
  equal to the following divisor
\begin{equation}
K_{\widetilde{C}} - v^*K_C = \sum_{i=1}^k \lt( a(E_i;B,r\cdot B_{g-1}) - 1 \rt)
\widetilde{\pi}^* E_i.
\end{equation}
\end{lem}

\begin{proof}
By the isomorphism in the proof of Lemma~\ref{lem-logres}, $u^{-1}\mc{I} \cong
\text{det} (\widetilde{\mc{E}}) \otimes_{\OO_{\widetilde{B}}} u^*
\text{det}(\mc{E})^\vee$.  On the other hand, applying
Lemma~\ref{lem-discrep} to both $C$ and $\widetilde{C}$, 
\begin{equation}
K_{\widetilde{C}} - v^* K_C = \widetilde{\pi}^* \lt( K_{\widetilde{B}} -
u^* K_B + r\cdot C_1(\text{det}(\widetilde{\mc{E}}) ) -
r\cdot u^*C_1(\text{det}(\mc{E}) ) \rt).
\end{equation}
\end{proof}



\medskip\noindent
\begin{prop} \label{prop-relcan} 
The pair $(C,\emptyset)$ is log canonical (resp. Kawamata log
terminal, canonical) iff the pair $(B,r\cdot B_{g-1})$ is log
canonical (resp. Kawamata log terminal, canonical).
\end{prop}

\medskip\noindent
\begin{proof}
Of course the total discrepancy of $(C,\emptyset)$,
$\text{totaldiscrep}(C,\emptyset)$, is the minimum of $0$ and the
discrepancy of $C$, $\text{discrep}(C,\emptyset)$.  So $(C,\emptyset)$
is log canonical (resp. Kawamata log terminal, canonical) iff
$\text{totaldiscrep}(C,\emptyset)$ is $=0$ (resp. $> -1$, $\geq -1$).
By a standard argument, cf. ~\cite[Cor. 2.32]{KM} and
~\cite[Prop. 1.3(iv)]{EMY}, $\text{totaldiscrep}(C,\emptyset) =
\text{totaldiscrep}(\widetilde{C}, -K_{\widetilde{C}/C})$.  Since
$\widetilde{C}$ is smooth and $\widetilde{\pi}^*(E_1 \cup \dots \cup
E_k)$ is a simple normal crossings divisor, the total discrepancy is
given by a combinatorial formula in the coefficients
$a(\widetilde{\pi}^*(E_i); \widetilde{C}, - K_{\widetilde{C}/C})$,
cf. ~\cite[Defn. 2.28]{KM} and ~\cite[Defn.  2.34]{KM}. 
But these are the same as the
coefficients $a(E_i;B,r\cdot B_{g-1})$.  So the total discrepancy of
$(C,\emptyset)$ equals the minimum of $0$ and the integers $a(E_i;B,
r\cdot B_{g-1})-1$.  Moreover, by assumption all of the divisors $E_i$
are exceptional for $u$.  Therefore the minimum of the integers
$a(E_i;B, r\cdot B_{g-1})-1$ is the discrepancy of $(B, r\cdot
B_{g-1})$.  Therefore $(C,\emptyset)$ is log canonical, etc. iff
$(B,r\cdot B_{g-1})$ is log canonical, etc.
\end{proof}

\medskip\noindent
\begin{rmk} \label{rmk-res} 
If each log discrepancy $a(E_i;B,r\cdot B_{g-1})$ is different than
$1$, then the exceptional locus of $v:\widetilde{C}
\rightarrow C$ is all of $\widetilde{\pi}^{-1}(E_1 \cup \dots \cup
E_k)$ and $v$ is a strong resolution of singularities.  In this case
$C$ is terminal iff $(B,r\cdot B_{g-1})$ is
terminal.  But if any log discrepancy equals $1$, this can fail;
e.g. if $B=\AAA^2$ and $\phi:\OO_{\AAA^2} \rightarrow
\OO_{\AAA^2}^{\oplus 2}$ is the map with matrix $(x,y)^\dagger$, then
for $r=1$ the cone $C$ is the blowing up of $\AAA^2$ in the origin.
So $C$ is smooth, and thus terminal.  But the pair $(\AAA^2,\{0\})$ is
only canonical.
\end{rmk}

\medskip\noindent
\subsection{Further manipulations} \label{subsec-further}

\medskip\noindent 
The results of this subsection are straightforward and are not 
actually used in
the rest of the paper, but it is natural to state them here.  In this
subsection the Grassmannian cone $C$ is denoted $C_r$ to
emphasize the integer $r$.  Proofs are left to the reader.

\medskip\noindent
\begin{lem} \label{lem-further1} 
   Let $1\leq s < r \leq e$.  If $C_r$ is normal of pure dimension
  $b+r(e-r)$, then also $C_s$ is normal of pure dimension $b+s(e-s)$.
  Moreover, if $C_r$ is log canonical (resp. Kawamata log terminal,
  canonical, terminal) then also $C_s$ is log canonical
  (resp. Kawamata log terminal, canonical, terminal).
\end{lem}

\medskip\noindent 
Let $\phi^{(i)}:\mc{G}^{(i)} \rightarrow \mc{F}^{(i)}$, $i=1,\dots,N$
be a sequence of injective morphisms of locally free sheaves such that
for each $i=1,\dots, N$ the cokernel $\mc{E}^{(i)}$ has generic rank
$e^{(i)} = f^{(i)} - g^{(i)}$.  Let $r^{(1)},\dots,r^{(N)}$ be a
sequence of integers with $1\leq r^{(i)} \leq e^{(i)}$.  Let
$\pi^{(i)}:C^{(i)} \rightarrow B$ denote the Grassmannian cone of rank
$r^{(i)}$ locally free quotients of $\mc{E}^{(i)}$ and let $\pi:C
\rightarrow B$ denote the fiber product $C^{(1)} \times_B \dots
\times_B C^{(N)}$.

\medskip\noindent
\begin{lem} \label{lem-further2} 
Suppose that $C$ has pure dimension $c=b+\sum_i
  r^{(i)}(e^{(i)}-r^{(i)})$.  Then Lemma~\ref{lem-discrep} applies to
  $\pi:C \rightarrow B$ where now the dualizing complex is
  $\omega_\pi[\sum_i r^{(i)}(e^{(i)}-r^{(i)})]$ with $\omega_\pi$
  equal to the tensor product of $\pi^*\lt[
  \text{det}(E^{(1)})^{\otimes r^{(1)}} \otimes \dots \otimes
  \text{det}(E^{(N)})^{\otimes r^{(N)}} \rt]$ with
  $\OO_{C^{(1)}}(-e^{(1)})\otimes \dots \otimes
  \OO_{C^{(N)}}(-e^{(N)})$.
\end{lem}

\medskip\noindent
\begin{lem} \label{lem-further3} 
Suppose that $C$ is normal of pure dimension $c$.  
For each
  $i=1,\dots,N$ let $Z^{(i)}$ denote the closed subscheme associated
  to the $e^{(i)}$ Fitting ideal of $\phi^{(i)}$.  
\begin{enumerate}
\item[(i)]
For each
  divisor $E$ of $K(B)$, the log discrepancy
  $a(\pi^{-1}(E);C,\emptyset)$ equals the log discrepancy
  $a(E;B,\sum_i r^{(i)} Z^{(i)})$.  
\item[(ii)]
The pair $(C,\emptyset)$ is log
  canonical (resp. Kawamata log terminal, canonical) iff $(B,\sum_i
  r^{(i)} Z^{(i)})$ is log canonical (resp. Kawamata log terminal,
  canonical).  
\item[(iii)]
For every subset $I \subset \{1, \dots, N \}$,
  the fiber product $C^{(I)} = \prod_B (C^{(i)} | i \in I )$ is normal
  of pure dimension $b + \sum_{i\in I} r^{(i)} Z^{(i)}$.
\item[(iv)] 
If $C$ is
  log canonical (resp. Kawamata log terminal, canonical), then
  $C^{(I)}$ is log canonical (resp. Kawamata log terminal, canonical).
\end{enumerate}
\end{lem}

\medskip\noindent
Define $\mc{G}:=\oplus_i \mc{G}_i$, define $\mc{F}:= \oplus_i
\mc{F}_i$, define $\phi:\mc{G} \rightarrow \mc{F}$ to be the direct
sum over $i=1,\dots, N$ of $\phi^{(i)}$ and define $\mc{E}$ to be the
cokernel of $\phi$, i.e. $\mc{E} \cong \oplus_i \mc{E}^{(i)}$.  Denote
by $e$ the sum $\sum_i e^{(i)}$ and let $r$ be an integer $1\leq r
\leq e$.  Define $\pi':C' \rightarrow B$ to be the Grassmannian cone
parametrizing rank $r$ locally free quotients of $\mc{E}$.


\medskip\noindent
\begin{lem} \label{lem-further4} 
\begin{enumerate}
\item[(i)] 
The $e^{\text{th}}$ Fitting ideal of $\phi$ is the product
  $\mc{I}^{(1)}\cdot \dots \cdot \mc{I}^{(N)}$.
\item[(ii)]  
If $C'$ is normal of pure dimension $b+r(e-r)$, then for every
  subset $I \subset \{1,\dots,N\}$, the Grassmannian bundle $C'_I
  \rightarrow B$ parametrizing rank $r$ locally free quotients of
  $\oplus_{i\in I} \mc{E}^{(i)}$ is normal of pure dimension
  $b+r(\sum_{i\in I} e^{(i)} -r)$. 
\item[(iii)] 
If, moreover, $C'$ is log canonical (resp. Kawamta log terminal,
  canonical), then also $C'_I$ is log canonical
  (resp. Kawamata log terminal, canonical).
\end{enumerate}
\end{lem}

\section{Log discrepancies of generic determinantal varieties}
\label{sec-deter} 

\medskip\noindent
Let $K$ be a field, not necessarily algebraically closed nor of
characteristic zero.  In this section all schemes
are $K$-schemes.  The interested reader will see how to
prove analogous results over $\SP(\ZZ)$, and thus over an
arbitrary base scheme.

\medskip\noindent
Let $S$ be a $K$-scheme and let $\mc{G}$, $\mc{F}$ be locally free
$\OO_S$-modules of finite rank $g$ and $f$ respectively with $g\leq
f$.  

\begin{notat} \label{notat-matrices}
Denote by $\pi^{(0)}:M^{(0)}(S,\mc{G},\mc{F})\rightarrow S$ 
the affine bundle $\underline{\text{Spec}}_S
\text{Sym}^*\lt(\textit{Hom}_{\OO_S}(\mc{G},\mc{F})^\vee \rt)$.  When
there is no risk of confusion, denote $M^{(0)}(S,\mc{G},\mc{F})$ by
$M^{(0)}$.  There is a tautological sheaf map, denoted
\begin{equation}
\phi: \mc{G}\otimes_{\OO_S} \OO_{M^{(0)}} \rightarrow \mc{F}
\otimes_{\OO_S} \OO_{M^{(0)}}.
\end{equation}
\end{notat}

\medskip\noindent
\begin{defn} \label{defn-gdv} 
For $k=0,\dots, g$, the $k^{\text{th}}$ \emph{generic determinantal
variety} $M^{(0)}_k \subset M^{(0)}$ is defined to be the closed
subscheme of $M^{(0)}$ whose ideal sheaf is generated by the
$(k+1)\times (k+1)$-minors of $\phi$ just as in
Notation~\ref{notat-Bk} (c.f. also ~\cite[Sec. II.2]{ACGH}).  For
technical reasons, denote by $M^{(0)}_{-1}$ the empty set.  In
particular, $M^{(0)}_0$ is just the zero section of $\pi^{(0)}:M^{(0)}
\rightarrow S$.
\end{defn}

\medskip\noindent
In this section the log discrepancies of the pair
$(M^{(0)}, M^{(0)}_k)$ are computed.  
This is straightforward once a log
resolution is constructed.  The log resolution is constructed 
by first blowing up
$M^{(0)}_0$, then blowing up the strict transform of $M^{(0)}_1$, the
blowing up the strict transform of $M^{(0)}_2$, etc.

\subsection{The log resolution} \label{subsec-logres} 
\medskip\noindent 
The log resolution of $(M^{(0)}, M^{(0)}_k)$ used here is the
obvious one: succesively blow up the strict transforms of the
schemes $M^{(0)}_0, M^{(0)}_1,\dots, M^{(0)}_k$.  Using the action of
the group $\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G})$, it is easy
to prove this does give a log resolution.  For completeness, we go
through the proof in (somewhat tedious) detail.

\medskip\noindent
\begin{rmk} \label{rmk-wonderful} 
The set of subschemes $M^{(0)}_k \subset M^{(0)}$ form a
stratification, and in case the ground field is $\CC$, this is a
\emph{conical stratification} in the sense of ~\cite{MP98} and the
blowing up we construct coincides with the \emph{minimal wonderful
compactification}.  We choose not to follow ~\cite{MP98} for two
reasons: First of all, the log resolution is quite general
and is valid over an arbitrary ground field $K$, 
not just over $\CC$ (and in fact over $\SP(\ZZ)$,
though we don't prove this).  More importantly, in computing log
discrepancies later, it is
\emph{crucial} that the log resolution has the additional
property that for each $M^{(0)}_k$, the inverse image of the ideal
sheaf of $M^{(0)}_k$ is an invertible sheaf, i.e., it contains no
embedded points.  This typically fails for the minimal wonderful
compactification associated to a conical stratification.  For example, 
consider the nodal plane cubic $p\in C$ sitting in $\PP^2$ sitting as
a linear subvariety of $\PP^3$.  Then $(\{p\}, C - \{p\}, \PP^3 - C)$
is a conical stratification of $\PP^3$.  And the inverse image of the
ideal sheaf of $C$ in the minimal wonderful compactification has an
``embedded line'' on the exceptional divisor over $p$.  It would be
interesting to know if there are extra hypotheses of a general nature
that can be added to the definition of a conical stratification so
that the minimal wonderful compactification has the additional
property.
\end{rmk}

\medskip\noindent
In the case that $f=g$, the log resolution in this section is identical
to that in ~\cite{Kausz}.  Moreover in ~\cite{Kausz} it is proved that
the inverse image of the ideal sheaf of $M^{(0)}_k$ is an invertible
sheaf.  However the case of most interest here is $f\neq g$.  Thus
the full description of the log resolution and proofs of the
basic properties of the log resolution are given.  The next lemma
gives a
precise definition of the sequence of blowing ups mentioned above.

\medskip\noindent
\begin{lem} \label{lem-exist} 
There exists a sequence of schemes $M^{(r)}$ for $r=0,\dots,g$ and
morphisms $u^{(s,r)}:M^{(r)} \rightarrow M^{(s)}$ for each $0\leq s
\leq r \leq g$ with the following properties
\begin{enumerate}
\item[(i)] 
For $0\leq t \leq s \leq r \leq g$, we have $u^{(t,s)}\circ
  u^{(s,r)} = u^{(t,r)}$.
\item[(ii)] 
For $r=0,\dots, g$, the morphism $u^{(0,r)}:M^{(r)}\rightarrow
  M^{(0)}$ is an isomorphism over the open subscheme $M^{(0)} -
  M^{(0)}_{r-1}$.
\item[(iii)] 
For each $0\leq r \leq k \leq g$, define $M^{(r)}_k \subset
  M^{(r)}$ to be the closure of the pullback by $u^{(0,r)}$ of
  $M^{(0)}_r - M^{(0)}_{k-1}$.  Then, for $r=0,\dots,g-1$ the morphism
  $u^{(r,r+1)}:M^{(r+1)} \rightarrow M^{(r)}$ is the blowing up of
  $M^{(r)}$ along $M^{(r)}_r$.
\end{enumerate}
\end{lem}
  
\begin{proof}  
This is almost tautological.  The one thing that needs to be checked
is that, defining $M^{(r+1)}$ to be the blowing up of $M^{(r)}$
along $M^{(r)}_r$, the induced map $u^{0,r+1}:M^{(r+1)}
\rightarrow M^{(0)}$ is an isomorphism over $M^{(0)} - M^{(0)}_{r}$.
But this follows immediately from the two facts: $u^{r,0}$ is an
isomorphism over $M^{(0)} - M^{(0)}_{r-1}$ and 
$u^{r,r+1}$ is an isomorphism over the preimage under $u^{r,0}$ of
$M^{(0)} - M^{(0)}_r$.
\end{proof}

\medskip\noindent
\begin{notat} \label{notat-Erk} 
Let $(S,\mc{G},\mc{F})$ be a datum with $\text{rank}(\mc{G}) =g$,
$\text{rank}(\mc{F})=f$ (and of course $g\leq f$). For each
$r=1,\dots,k$, denote by $E^{(r)}_{r-1}(S,\mc{G},\mc{F}) \subset
M^{(r)}(S,\mc{G},\mc{F})$ the exceptional divisor of the blowing up
$u^{r-1,r}$.  For $r=2,\dots,k$, and for $k=0,\dots,r-2$, denote by
$E^{(r)}_k\subset M^{(r)}$ the strict transform of $E^{(k+1)}_k$ under
the morphism $u^{k+1,r}:M^{(r)} \rightarrow M^{(k+1)}$.  Clearly for
each $0\leq r< s \leq g$, the exceptional locus of $u^{r,s}:M^{(s)}
\rightarrow M^{(r)}$ is $E^{(s)}_{r} \cup \dots \cup E^{(s)}_{s-1}$.
\end{notat}

\medskip\noindent
\begin{defn} \label{defn-mor} 
Let $(S,\mc{G}',\mc{F}')$ and $(S,\mc{G},\mc{F})$ be data of pairs of
locally free sheaves on $S$ with $g' \leq f'$ and $g\leq f$.  A
\emph{morphism} between the data is a triple $\zeta = (p,q,T)$ where
\begin{enumerate}
\item[(i)]
$p:\mc{G} \rightarrow \mc{G}'$ is a surjective morphism of
$\OO_S$-modules,
\item[(ii)] 
$q:\mc{F}'\rightarrow \mc{F}$ is an injective morphism of
$\OO_S$-modules whose cokernel is locally free, and
\item[(iii)]
$T:\mc{G} \rightarrow \mc{F}$ is a morphism of $\OO_S$-modules,
\end{enumerate}
and
such that there are direct sum decompositions 
\begin{equation}
\mc{G} = \text{Ker}(T) \oplus \text{Ker}(p),\ \ \mc{F} =
\text{Image}(T) \oplus \text{Image}(q).
\end{equation}
In particular, $g-g' = f-f'$.  The \emph{rank} of $\zeta$ is
the common integer $g-g'=f-f'$.
\end{defn}

\medskip\noindent
\begin{lem} \label{lem-mor} 
Let $\zeta = (p,q,T)$ be a morphism $(S,\mc{G}',\mc{F}') \rightarrow
(S,\mc{G},\mc{F})$ of rank $l$.  For $r=0,\dots,g'$ there
exist morphisms of $S$-schemes
\begin{equation}
\tau^{r}(\zeta):\text{GL}(\mc{F}) \times_S
\text{GL}(\mc{G}) \times_S M^{(r)}(S,\mc{G}',\mc{F}') \rightarrow
M^{(r+l)}(S,\mc{G},\mc{F}) 
\end{equation}
satisfying the following conditions.
\begin{enumerate}
\item[(i)] 
The image of the composite morphism $u^{0,l}\circ \tau^0(\zeta)$ 
is contained in $M^{(0)}(S,\mc{G},\mc{F}) -
M^{(0)}_{l-1}(S,\mc{G},\mc{F})$.
\item[(ii)] 
The morphism $\tau^0(\zeta)$ is the unique morphism such that 
$u^{0,l}\circ \tau^0(\zeta)$ is the morphism whose
restriction to the fiber over a point $x\in S$ maps a triple
$(\alpha,\beta,L)$ in $\text{GL}(\mc{F}_x) \times \text{GL}(\mc{G}_x)
\times M^{(0)}(x,\mc{G}'_x, \mc{F}'_x)$ to the element $\alpha\circ
(T_x + q_x\circ L \circ p_x)\circ \beta^{-1}$.
\item[(iii)]
For $0\leq r \leq s \leq g'$, $u^{l+r,l+s}\circ
\tau^{s}(\zeta)$ equals $\tau^r(\zeta) \circ \lt(\text{Id} \times
\text{Id} \times u^{r,s}\rt)$; moreover, the corresponding
commutative diagram is Cartesian.
\item[(iv)] 
For each $0\leq r \leq g'$, the morphism $\tau^r(\zeta)$ is
quasi-compact, separated and smooth, and 
$\text{Image}(\tau^r(\zeta)) = (u^{0,l+r})^{-1}(M^{(0)} - M^{(0)}_{l-1})$.
\item[(v)] 
For each $0\leq r\leq k \leq g'$, 
\begin{equation*}
(\tau^r(\zeta))^{-1}(M^{(r+1)}_{k+1}(S,\mc{G},\mc{F})) = 
\text{GL}(\mc{F})\times_S \text{GL}(\mc{G}) \times_S
M^{(r)}_k(S,\mc{G}',\mc{F}').
\end{equation*} 
\item [(vi)]
For each $0\leq r \leq g'$ and each $0\leq i \leq r-1$,
\begin{equation*}
(\tau^r(\zeta))^{-1}(E^{(r+l)}_{i+l}(S,\mc{G},\mc{F}) =
\text{GL}(\mc{F})\times_S \text{GL}(\mc{G}) \times_S
E^{(r)}_{i}(S,\mc{G}',\mc{F}'). 
\end{equation*}
In particular, this is a Cartier divisor.
\end{enumerate}
\end{lem}

\medskip\noindent
\begin{proof}
The claim is that for each $0 \leq r \leq g'$, there exists a sequence
of morphisms $\tau^0(\zeta), \dots, \tau^r(\zeta)$ satisfying
(i)--(vi).
The claim is proved by induction on $r$.  First consider the case
$r=0$.
The morphism in (ii),
\begin{equation}
\tau(\zeta): \text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S
M^{(0)}(S,\mc{G}',\mc{F}') \rightarrow M^{(0)}(S,\mc{G},\mc{F}),
\end{equation}
is defined more precisely
by giving a natural transformation of the obvious functors represented
by the two schemes and invoking Yoneda's lemma.

\medskip\noindent
Let $T$ be a $k$-scheme.  By the universal properties of the
three factors, a morphism from $T$ to $\text{GL}(\mc{F}) \times_S
\text{GL}(\mc{G}) \times_S M^{(0)}(S,\mc{G}',\mc{F}')$ is equivalent
to a morphism $f:T\rightarrow S$ together with a triple
$(\alpha,\beta,L)$ consisting of $\alpha:f^*\mc{F} \rightarrow f^*\mc{F}$ 
an automorphism of $\OO_T$-modules, $\beta:f^*\mc{G} \rightarrow
f^*\mc{G}$ is an automorphism of $\OO_T$-modules, and 
$L:f^*\mc{G}' \rightarrow f^*\mc{F}'$ a morphism of
$\OO_T$-modules.  There is an associated morphism of $\OO_T$-modules
$f^*\mc{G} \rightarrow f^*\mc{F}$ by $L'=\alpha\circ \lt( f^*T +
f^*q\circ L f^*p\rt) \circ \beta^{-1}$.  By the universal property of
$M^{(0)}(S,\mc{G},\mc{F})$ the pair $(f,L')$ is equivalent to a
morphism $T\rightarrow M^{(0)}(S,\mc{G},\mc{F})$.  The association
$(f,\alpha,\beta,L)\mapsto (f,L')$ is a natural transformation
of Yoneda functors and so determines a morphism of $S$-schemes
$\tau$.  

\medskip\noindent
\textbf{(i) and (ii)}:
Using the direct sum decompositions of $\mc{G}$ and $\mc{F}$
induced by $(p,q,T)$, for any point $x\in T$ the rank of
$L'_x$ equals $\text{rank}(T_{f(x)}) + \text{rank}(L_x)$,
i.e. $l+\text{rank}(L_x)$. This is bigger than $l-1$, hence the
image of $\tau$ is contained in the complement of
$M^{(0)}_{l-1}(S,\mc{G},\mc{F})$.  Since
$u^{0,l}:M^{(l)}(S,\mc{G},\mc{F}) \rightarrow
M^{(0)}(S,\mc{G},\mc{F})$ is an isomorphism over the complement of
$M^{(0)}_{l-1}(S,\mc{G},\mc{F})$, there is a unique morphism
$\tau^0(\zeta)$ such that $u^{0,l}\circ \tau^0(\zeta)$ equals $\tau$.

\medskip\noindent
\textbf{(iv); r=0}:
This is equivalent
to the claim that $\tau$ is quasi-compact, smooth and separated.  It
is clear that both $\text{GL}(\mc{F})\times_S \text{GL}(\mc{G})
\times_S M^{(0)}(S,\mc{G}',\mc{F}')$ and $M^{(0)}(S,\mc{G},\mc{F})$
are quasi-compact, smooth and separated over $S$.  Therefore $\tau$ is
quasi-compact, finitely-presented and separated.  To show that
$\tau$ is smooth it suffices to check the Jacobian criterion for
fibers of $\tau$ over geometric points of $S$.

\medskip\noindent
Let $x\in S$ be a geometric point and let $(\alpha,\beta,L)$
be a closed point of the fiber of
$\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S
M^{(0)}(S,\mc{G}',\mc{F}')$ over $x$.  The smooth group
scheme $\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G})$ acts on both the
domain and target of $\tau$ and $\tau$ is equivariant for this action.
Therefore it suffices to check the Jacobian criterion at one
representative point of every orbit.  Hence it suffices to consider
the special case
$\alpha=\text{id}_{\mc{F}}$ and $\beta = \text{id}_{\mc{G}}$.
The Zariski tangent space to the fiber of $\text{GL}(\mc{F}) \times_S
\text{GL}(\mc{G}) \times_S M^{(0)}(S,\mc{G}',\mc{F}')$ over $x$ at any 
closed point is
canonically identified with the $\kappa(x)$-vector space
$\text{Hom}(\mc{F}_x,\mc{F}_x) \times \text{Hom}(\mc{G}_x,\mc{G}_x)
\times \text{Hom}(\mc{G}'_x,\mc{F}'_x)$.  Similarly the Zariski
tangent space to the fiber of $M^{(0)}(S,\mc{G},\mc{F})$ is
canonically identified with $\text{Hom}(\mc{G}_x,\mc{F}_x)$.  And
$d\tau$ maps a triple $(\alpha_1, \beta_1, L_1)$ to the element
$\alpha_1\circ(T_x+q_x\circ L\circ p_x) + q_x\circ L_1 \circ p_x +
(T_x+q_x\circ L\circ p_x) \circ \beta_1$.

\medskip\noindent
Let $d=\text{rank}(L)$.  There exists an ordered bases for $\mc{G}_x$
and $\mc{F}_x$ with respect to which $T_x$ has the matrix
representation,
\begin{equation}
\lt[ \begin{array}{r|r|r}
I_{l,l} & 0_{l,d} & 0_{l,g'-d} \\
\hline 0_{d,l} & 0_{d,d} & 0_{d,g'-d} \\
\hline 0_{f'-d,l} & 0_{f'-d,d} & 0_{f'-d,g'-d}
\end{array} \rt],
\end{equation}
and with respect to which $q_x\circ L \circ p_x$ has the matrix
representation,
\begin{equation}
\lt[ \begin{array}{r|r|r}
0_{l,l} & 0_{l,d} & 0_{l,g'-d} \\
\hline 0_{d,l} & I_{d,d} & 0_{d,g'-d} \\
\hline 0_{f'-d,l} & 0_{f'-d,d} & 0_{f'-d,g'-d}
\end{array} \rt].
\end{equation}
For any linear operator $L'\in \text{Hom}(\mc{G}_x,\mc{F}_x)$ consider
the matrix representation of $L'$ with respect to the ordered bases
above,
\begin{equation}
\lt[ \begin{array}{r|r|r}
L'_1 & L'_2 & L'_3 \\
\hline L'_4 & L'_5 & L'_6 \\
\hline L'_7 & L'_8 & L'_9
\end{array} \rt],
\end{equation}
where the block submatrices $L'_i$ have the same dimensions as the
blocks in the matrices of $T$ and $L$.  Then, denoting,
\begin{equation}
\alpha_1 = \lt[ \begin{array}{r|r|r}
L'_1 & L'_2 & 0 \\
\hline L'_4 & L'_5 & 0 \\
\hline L'_7 & L'_8 & 0 
\end{array}\rt],
\end{equation}
\begin{equation}
\beta_1 = \lt[ \begin{array}{r|r|r}
0_{l,l} & 0_{l,d} & L'_3 \\
\hline 0_{d,l} & 0_{d,d} & L'_6 \\
\hline 0_{f'-d,l} & 0_{f'-d,d} & 0_{f'-d,g'-d}
\end{array} \rt],
\end{equation}
\begin{equation}
q_x\circ L_1 \circ p_x = \lt[ \begin{array}{r|r|r}
0_{l,l} & 0_{l,d} & 0_{l,g'-d} \\
\hline 0_{d,l} & 0_{d,d} & 0_{d,g'-d} \\
\hline 0_{f'-d,l} & 0_{f'-d,d} & L'_9
\end{array} \rt],
\end{equation}
the pair $(\alpha_1,\beta_1,L_1)$ maps to $L'$ under
$d\tau$.  Hence $\tau$ satisfies the Jacobian criterion and $\tau$
and $\tau^{0}(\zeta)$ are smooth.  

\medskip\noindent
\textbf{(iii), (v), and (vi); r=0}:
For $L' = \tau(\alpha,\beta,L)$, $\text{rank}(L') = l +
\text{rank}(L)$. Hence the preimage under $\tau$ of
$M^{(0)}_{l+k}(S,\mc{G},\mc{F})$ is the closed subscheme
$\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S M^{(0)}_{k}(S,
\mc{G}', \mc{F}')$; i.e., (v) holds. 
 Finally (iii) and (vi) are vacuous for
the ``sequence'' of morphisms $\tau^0(\zeta)$.  This finishes the base
case $r=0$.

\medskip\noindent
\textbf{(iii) and (iv); induction step}:
Now comes the induction step.  Let $r=1,\dots,g'$.  By way of
induction,
suppose that
morphisms $\tau^{0}(\zeta),\dots,\tau^{r-1}(\zeta)$ have been
constructed satisfying (i)--(vi).  
Since $\tau^{r-1}(\zeta)$ is
smooth, the fiber product of $\tau^{r-1}(\zeta)$ with the blowing up
$u^{l+r-1,l+r}$
%M^{(l+r)}(S,\mc{G},\mc{F}) \rightarrow
%M^{(l+r-1)}(S,\mc{G},\mc{F})$ of $M^{(l+r-1)}(S,\mc{G},\mc{F})$ along
%the closed subscheme $M^{(l+r-1)}_{l+r-1}(S,\mc{G},\mc{F})$ 
is
canonically isomorphic to the blowing up of $\text{GL}(\mc{F})\times_S
\text{GL}(\mc{G})\times_S M^{(r-1)}(S,\mc{G}',\mc{F}')$ along the
preimage of $M^{(l+r-1)}_{l+r-1}(S,\mc{G},\mc{F})$.  By the induction
hypothesis, the preimage is precisely $\text{GL}(\mc{F}) \times_S
\text{GL}(\mc{G}) \times_S M^{(r-1)}_{r-1}(S,\mc{G}',\mc{F}')$.  So
the base-change of $u^{l+r-1,l+r}$ by $\tau^{r-1}(\zeta)$ is just
$\text{Id} \times \text{Id}\times u^{r-1,r}$.  Define
$\tau^r(\zeta):\text{GL}(\mc{F})\times_S \text{GL}(\mc{G}) \times_S
M^{(r)}(S,\mc{G}',\mc{F}') \rightarrow M^{(l+r)}(S,\mc{G},\mc{F})$ to
be the base-change of $\tau^{r-1}(\zeta)$ by $u^{l+r-1,l+r}$.

\medskip\noindent
By construction of $\tau^r(\zeta)$ and the induction hypothesis, 
$\tau^0(\zeta),\dots,\tau^r(\zeta)$ satisfies (iii).  
Since $\tau^r(\zeta)$ is the base-change of a
quasi-compact, separated and smooth morphism, $\tau^r(\zeta)$ is also
quasi-compact, separated and smooth, i.e. (iv) holds.

\medskip\noindent
\textbf{(v); induction step}:
Since $\tau^r(\zeta)$ and
$\tau^{r-1}(\zeta)$ are smooth, and by (iii), the process of
forming the strict transform by $u^{l+r,l+r-1}$ of a closed subscheme
and then forming the preimage under $\tau^r(\zeta)$ is the same as the
process of first forming the preimage under $\tau^{r-1}(\zeta)$ and
then forming the strict transform under $\text{Id} \times \text{Id}
\times u^{r,r-1}$.  By the induction hypothesis and (v), the
preimage under $\tau^{r-1}(\zeta)$ of $M^{(r+l-1)}_k(S,\mc{G},\mc{F})$
equals $ \text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S
M^{(r-1)}_k(S,\mc{G}',\mc{F}')$.  The strict transform of this
subscheme under $\text{Id} \times \text{Id} \times u^{r-1,r}$ is
$\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S
M^{(r)}_k(S,\mc{G}',\mc{F}')$.  Therefore (v) is satisfied for
$\tau^r(\zeta)$.

\medskip\noindent
\textbf{(vi); induction step}:
As above, for
$i=0,\dots,r-2$ the pullback by $\tau^r(\zeta)$ of the strict
transform by $u^{l+r,l+r-1}$ of $E^{(l+r-1)}_{i+l}(S,\mc{G},\mc{F})$
equals the strict transform of the pullback by $\tau^{r-1}(\zeta)$.
By the induction hypothesis and (vi), this pullback is
$\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S
E^{(r-1)}_i(S,\mc{G}',\mc{F}')$.  The strict transform of this
subscheme under $\text{Id} \times \text{Id} \times u^{r,r-1}$ is
$\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S
E^{(r)}_i(S,\mc{G}',\mc{F}')$.  Finally, using (iii), the
preimage under $\tau^r(\zeta)$ of the exceptional divisor of
$u^{r+l,r+l-1}$, i.e. of $E^{(r+l)}_{r+l-1}(S,\mc{G},\mc{F})$, equals the
exceptional divisor of $\text{Id} \times \text{Id} \times u^{r,r-1}$,
i.e. $\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S
E^{(r)}_{r-1}(S,\mc{G}', \mc{F}')$.  This proves (vi) and
finishes the induction step.  The lemma is proved by induction on $r$.
\end{proof}

\medskip\noindent
\begin{lem} \label{lem-snc} 
Let $V$ be a smooth $K$-scheme and let $D\subset V$ be a simple normal
crossings divisor.  The $K$-scheme $\AAA^1 \times V$ is smooth and the
divisor $D' = (\AAA^1\times D) \cup (\{0\} \times V)$ is a simple
normal crossings divisor in $\AAA^1 \times V$.
\end{lem}

\medskip\noindent
\begin{proof}  This follows immediately from the definition of
  \emph{simple normal crossings divisor}.
\end{proof}

\medskip\noindent
\begin{notat} \label{notat-Ur} 
Denote by $U^{(0)} \subset M^{(0)}(S,\mc{G},\mc{F})$ the complement of
the zero section $M^{(0)}_0(S,\mc{G},\mc{F})$.  For each $r=0,\dots,
g$ denote by $U^{(r)} \subset M^{(r)}$ the open subscheme $U^{(r)} =
\lt(u^{0,r}\rt)^{-1}(U^{(0)})$ and denote by $v^{r,s}:U^{(s)}
\rightarrow U^{(r)}$ the restriction of $u^{r,s}$ to $U^{(s)}$.  For
each $r=0,\dots,g$ and each $i=r,\dots,g$, denote by $\mc{I}^{(r)}_k$
the ideal sheaf of the closed subscheme $M^{(r)}_k \subset M^{(r)}$.
Finally denote by $f:\AAA^1 \times U^{(0)} \rightarrow M^{(0)}$ the
morphism sending a pair $(\lambda, L)$ in $\AAA^1 \times U^{(0)}$
to the point $\lambda\cdot L \in M^{(0)}$.
\end{notat}  

\medskip\noindent
The preimage under $f$ of $M^{(0)}_0$ is precisely $\{0\} \times
U^{(0)}$, which is a Cartier divisor in $\AAA^1 \times U^{(0)}$.
Therefore, by the universal property of blowing up, there is a unique
morphism $f^{1}:\AAA^1 \times U^{(0)} \rightarrow M^{(1)}$ such that
$u^{0,1}\circ f^{1} = f$.  It is easy to check that $f^{1}:\AAA^1
\times U^{(1)} \rightarrow M^{(1)}$ is a $\mathbb{G}_m$-torsor where
$\mathbb{G}_m$ acts on $\AAA^1 \times U^{(1)}$ by $\mu\cdot
(\lambda,L) = (\mu\cdot \lambda, \mu^{-1}\cdot L)$.  In particular,
$f^{1}$ is smooth and surjective.  The preimage under $f^{1}$ of
$E^{(1)}_0$ is $\{0\} \times U^{(1)}$.  The next lemma proves that for
$k =1,\dots, g-1$, the preimage under $f^{1}$ of $M^{(1)}_k$ is
$\AAA^1 \times \lt(U^{(1)}\cap M^{(1)}_k \rt)$.

\medskip\noindent
\begin{lem} \label{lem-pref} 
Define $\mc{G}^{(1)}$ to be the locally free sheaf
$(u^{0,1})^*\mc{G}(E^{(1)}_0)$ and let $(u^{0,1})^*\mc{G} \rightarrow
\mc{G}^{(1)}$ denote the canonical sheaf map.
\begin{enumerate}
\item[(i)] 
There is a factorization $\phi^{(1)}:\mc{G}^{(1)} \rightarrow
(u^{0,1})^*\mc{F}$ of $(u^{0,1})^*\phi$.
\item[(ii)] 
The pullback of $\phi^{(1)}$ by $f^1$ is canonically isomorphic to the
pullback $\text{pr}_2^* \phi$ of the restriction of $\phi$ to $U^{(0)}
= U^{(1)}$.
\item[(iii)] 
For every geometric point $x$ of $M^{(1)}$,
$\text{rank}(\phi^{(1)}|_x) \geq 1$.
\item[(iv)] 
For each $k=1,\dots, g-1$, the inverse image ideal sheaf
$(u^{0,1})^{-1}(\mc{I}^{(0)}_k)$ equals the ideal sheaf
$\mc{I}^{(1)}_k\cdot
\OO_{M^{(1)}}( -(k+1)E^{(1)}_0 )$.
\item[(v)] 
For each $k=1,\dots, g-1$, the preimage under $f^{(1)}$ of $M^{(1)}_k$
equals $\AAA^1 \times (U^{(1)} \cap M^{(1)}_k )$.
\end{enumerate}
\end{lem}

\begin{proof}
\textbf{(i)}:
The restriction of $(u^{0,1})^*\phi$ to $E^{(1)}_0$ is the zero map.
Therefore it factors through the elementary-transform-up of
$(u^{0,1})^*\mc{G}$, i.e. it factors through a morphism
$\phi^{(1)}:\mc{G}^{(1)} \rightarrow (u^{0,1})^*\mc{F}$.  This
proves (i).

\medskip\noindent
\textbf{(iii)}:
The morphism $(f^1)^* \mc{G}^{(1)}$ equals $\text{pr}_2^*\mc{G}({0}\times
U^{(1)})$.  This is canonically isomorphic to $\text{pr}_2^*\mc{G}$.
Via this isomorphism, the pullback $(f^1)^* \phi^{(1)}$ equals
$\text{pr}_2^* \phi$.  Since $\phi|_{U^{(0)}}$ has rank at least $1$
at all geometric points, the same is true of $\text{pr}_2^* \phi$.
Therefore $\phi^{(1)}$ has rank at least $1$ at all geometric points.

\medskip\noindent
\textbf{(iv) and (v)}:
One can check that two ideal sheaves are equal after a faithfully flat
base change.  Since $f^1$ is faithfully flat, to prove
both (iv) and (v), it suffices to prove that the inverse
image ideal sheaf of $\mc{I}^{(0)}_k$ in $\AAA^1 \times U^{(1)}$
equals
\begin{equation}
\text{pr}_2^{-1}(\mc{I}^{(0)}) \cdot \OO_{\AAA^1 \times
  U^{(1)}}(-(k+1)\{0\} \times U^{(1)}).
\end{equation}
Let $t$ denote the coordinate on $\AAA^1$.  Then the preimage under
$f^1$ of $(u^{0,1})^*\phi$ is precisely the matrix $t\cdot
\text{pr}_2^*\phi$.  Therefore the ideal sheaf generated by the
$(k+1)\times (k+1)$-minors of this matrix is just $t^{k+1}$ times the
ideal generated by $(k+1) \times (k+1)$-minors of $\text{pr}_2^*\phi$,
i.e. the inverse image ideal sheaf of $\mc{I}^{(0)}_k$ under
$u^{0,1}\circ f^1$ is as above.  
\end{proof}

\medskip\noindent
\begin{lem} \label{lem-f} 
For $r=1,\dots,g$ there exist morphisms of $S$-schemes 
$$f^{r}:
\AAA^1 \times U^{(r)} \rightarrow M^{(r)}$$
satisfying the
following conditions.
\begin{enumerate}
\item[(i)] 
The image of the composite morphism $u^{0,1}\circ f^{1}$ is
contained in $U^{(0)}$.
\item[(ii)] 
The morphism $f^{1}$ is the unique morphism such that $u^{1,0}\circ
f^{1} = f$.
\item[(iii)] 
For $1\leq r \leq s \leq g$, $u^{r,s} \circ f^{s}$ equals $f^{r} \circ
\lt( \text{Id} \times v^{r,s} \rt)$; moreover, the corresponding
commutative diagram
is Cartesian.
\item[(iv)] 
For each $1 \leq r \leq g$, the morphism $f^r$ is a
$\mathbb{G}_m$-torsor, in particular it is surjective and smooth.
\item[(v)] 
For each $1 \leq r \leq k \leq g$, 
\begin{equation*}
(f^r)^{-1}(M^{(r)}_k) = \AAA^1 \times (U^{(r)} \cap M^{(r)}_k ).
\end{equation*}
\item[(vi)] 
For each $2 \leq r \leq g$ and $1 \leq k \leq r-1$, 
\begin{equation*}
(f^r)^{-1}(E^{(r)}_k) = \AAA^1 \times (U^{(r)} \cap E^{(r)}_k ).
\end{equation*}
And for each $1\leq r \leq g$, the preimage under $f^r$ of
$E^{(r)}_0$ is $\{0\} \times U^{(r)}$.
\end{enumerate}
\end{lem}

\begin{proof}
\textbf{(i), (ii), and $r=1$}:
Item (i) is trivial and is only included to maintain symmetry with
Lemma~\ref{lem-mor}.  Item (ii) follows from the construction of
$f^{(1)}$.  
The claim is that for each $1\leq r \leq g$, there exists a sequence
of morphisms $f^1,\dots,f^r$ satisfying (i)--(vi).  The claim is
proved by induction on $r$.  For $r=1$ this has already been
established; in particular, (v) follows from (v) of Lemma~\ref{lem-pref}.

\medskip\noindent
\textbf{(iii) and (iv); induction step}:
Now comes the induction step.  Let $r=2,\dots,g$.  By way of
induction, suppose that morphisms $f^1,\dots, f^{r-1}$ have been
constructed satisfying (i)--(vi).
Since $f^{r-1}$ is smooth, the fiber product of $f^{r-1}$
with the blowing up $u^{r-1,r}$
%M^{(r)} \rightarrow M^{(r-1)}$ along
%the closed subscheme $M^{(r-1)}_{r-1}$ 
is canonically isomorphic to
the blowing up of $\AAA^1 \times U^{(r-1)}$ along the preimage of
$M^{(r-1)}_{r-1}$.  By the induction hypothesis and (v), the
preimage is precisely $\AAA^1 \times (U^{(r-1)} \cap
M^{(r-1)}_{r-1})$.  So the base-change of $u^{r-1,r}$ by $f^{r-1}$ is
just $\text{Id} \times v^{r-1,r}$.  Define $f^r:\AAA^1 \times
U^{(r)} \rightarrow M^{(r)}$ to be the base-change of $f^{r-1}$ by
$u^{r-1,r}$.

\medskip\noindent
By construction of $f^r$ and the induction hypothesis, 
$f^1,\dots,f^r$ satisfies (iii).  Since $f^r$ is
the base-change of the $\mathbb{G}_m$-torsor $f^{r-1}$, also $f^r$ is
a $\mathbb{G}_m$-torsor, i.e. (iv) holds.

\medskip\noindent
\textbf{(v); induction step}:
Using the Cartesian property of (iii)
and since $f^{r-1}$ and $f^r$ are smooth, the process of forming
the strict transform under $u^{r-1,r}$ and then forming the preimage
under $f^r$ is the same as the process of first forming the preimage
under $f^{r-1}$ and then forming the strict transform under $\text{Id}
\times v^{r-1,r}$.  By the induction hypothesis and (v), the
preimage under $f^{r-1}$ of $M^{(r-1)}_k$ equals $\AAA^1 \times
(U^{(r-1)} \cap M^{(r-1)}_k)$.  The strict transform of this subscheme
under $\text{Id} \times v^{r-1,r}$ is $\AAA^1 \times (U^{(r)} \cap
M^{(r)}_k)$.

\medskip\noindent
\textbf{(vi); induction step}:
As above, for
$k=0,\dots,r-2$ the pullback by $f^r$ of the strict transform of
$E^{(r-1)}_k$ equals the strict transform of the pullback by
$f^{r-1}$.  By the induction hypothesis and (vi), the pullback
of $E^{(r-1)}_0$ equals $\{0\}\times U^{(r-1)}$ and the pullback of
$E^{(r-1)}_k$ equals $\AAA^1 \times (U^{(r-1)} \cap E^{(r-1)}_k)$ for
$k=1,\dots,r-2$.  The strict transforms of these subschemes are $\{0\}
\times U^{(r)}$ and $\AAA^1 \times (U^{(r)} \cap E^{(r)}_k)$
respectively.  Finally, using (iii), the preimage under $f^r$ of
the exceptional divisor of $u^{r-1,r}$, i.e. of $E^{(r)}_{r-1}$, equals
the exceptional divisor of $\text{Id} \times v^{r-1,r}$, i.e. $\AAA^1
\times (U^{(r)} \cap E^{(r)}_{r-1})$.  This proves (vi) and
finishes the induction step.  The lemma by induction on $r$.
\end{proof}

The next proposition is the main result of this section.

\medskip\noindent
\begin{prop} \label{prop-smooth} 
\begin{enumerate}
\item[(i)]
For each $r=0,\dots,g$, the scheme $M^{(r)}$ is smooth over $S$.  
\item[(ii)]
For each $r=1,\dots,g$, the closed subscheme $E^{(r)}_{0} \cup \dots \cup
E^{(r)}_{r-1}$ is a simple normal crossings divisor in $M^{(r)}$;
moreover the intersection with every geometric fiber over $S$ is a
simple normal crossings divisor.
\item[(iii)]
For each $r=1,\dots,g$, the scheme $M^{(r)}_r$ is smooth over $S$, and
therefore $M^{(r)}_r \rightarrow M^{(r)}$ is a regular embedding. 
\item[(iv)]
For $r=0,\dots,g$ there exist locally free sheaves of rank $g$ on $M^{(r)}$,
$\mc{G}^{(r)}$, and morphisms of sheaves
$\phi^{(r)}:\mc{G}^{(r)} \rightarrow (u^{0,r})^* \mc{F}$, such that
$\mc{G}^{(0)} = \mc{G}$ and $\phi^{(0)}=\phi$, such that $\mc{G}^{(1)}
= (u^{0,1})^* \mc{G}^{(0)}(E^{(1)}_0)$ and $\phi^{(1)}$ is as in Lemma
~\ref{lem-pref}, and that satisfy the following condition: for each
$r=1,\dots,g$ there is a factorization $(u^{r-1,r})^*\mc{G}^{(r-1)}
\xrightarrow{\psi^{(r)}} \mc{G}^{(r)} \xrightarrow{\phi^{(r)}}
(u^{0,r})^*\mc{F}$ such that the cokernel of $\psi^{(r)}$ is the
pushforward from $E^{(r)}_{r-1}$ of a locally free sheaf of rank
$g+1-r$ and such that $\phi^{(r)}$ has rank at least $r$ at all
geometric points, and has rank $g$ at the generic point of
$E^{(r)}_{r-1}$.
\item[(v)]  
The morphisms $\psi^{(r)}$ and $\phi^{(r)}$ above are unique up to
unique isomorphism (considering the sheaves $\mc{G}^{(r)}$ as 
subsheaves of $\mc{G}\otimes_{\OO_{M^{(0)}}} K(M^{(0)})$, then the
morphisms are honestly unique).  Moreover, they are equivariant for the
obvious action of the group scheme $\text{GL}(\mc{F}) \times
\text{GL}(\mc{G})$.
\item[(vi)]
For each $0 \leq r < s \leq g$ and each $k\geq s$, the inverse image
ideal sheaf $(u^{r,s})^*\mc{I}^{(r)}_k$ equals
\begin{equation}
\mc{I}^{(s)}_k \cdot
\OO_{M^{(s)}} ( - (\sum_{i=r}^{s-1} (k+1-i) E^{(s)}_i ) ).
\end{equation}
\end{enumerate}
\end{prop}

\medskip\noindent
\begin{proof}
\textbf{g=0}:
The proposition can be checked Zariski locally over $S$.  And Zariski
locally over $S$, the locally free sheaves $\mc{G}$ and $\mc{F}$ are
free.  Hence it suffices to consider the case that $S = \SP(K)$.  
The result is proved by induction on $g$.  For $g=0$, there is nothing to
prove.  Therefore, by way of induction, we may suppose that $g>0$ and
the result has been proved whenever $\text{rank}(\mc{G}) < g$.

\medskip\noindent
The idea of the induction step is described in the next 2 paragraphs.  
First of all, for
$M^{(0)}$ (i)--(vi) are obvious.  Thus it suffices
to consider $M^{(r)}$ with $r=1,\dots,g$.  The next step is to restrict
over $U^{(1)}, \dots, U^{(g)}$ and check (i)--(vi) through 
when restricted over these open sets.  This can be checked after
a smooth surjective base-change on the sets $U^{(r)}$.
Lemma~\ref{lem-mor} gives a sequence of such base-changes
$\tau^{r-1}(\zeta)$ for any rank $1$ morphism
$\zeta:(S,\mc{G}',\mc{F}') \rightarrow (S,\mc{G},\mc{F})$.  After
base-change by $\tau^{r-1}(\zeta)$, (i)--(vi) over
$U^{(r)}$ reduce to (i)--(vi) over
$M^{(r-1)}(S,\mc{G}',\mc{F}')$.  Since $\text{rank}(\mc{G}') <
\text{rank}(\mc{G})$, (i)--(vi) over $M^{(r-1)}(S,\mc{G}',\mc{F}')$
hold by the induction assumption.

\medskip\noindent
Next, to establish the proposition over all of $M^{(r)}$, consider the
sequence of morphisms $f^r:\AAA^1 \times U^{(r)} \rightarrow M^{(r)}$
from Lemma~\ref{lem-f}.  These morphisms are smooth and surjective, so
(i)--(vi) may be checked after base-change by $f^r$.
And these reduce to (i)--(vi) over $U^{(r)}$.  Since (i)--(vi) have
been proved over $U^{(r)}$,
this finishes the induction step.  The proof follows by induction.

\medskip\noindent
\textbf{(i); $U^{(r)}$}:
First (i)--(vi) are proved when restricted over
$U^{(1)},\dots , U^{(g)}$.  Let $T \in U^{(0)}$ be a geometric point
with $\text{rank}(T)=1$.  There exists $\mc{G}'$ of rank $g-1$,
$\mc{F}'$ of rank $f-1$, $p:\mc{G} \rightarrow \mc{G}'$, and
$q:\mc{F}' \rightarrow \mc{F}$ so that $\zeta=(p,q,T)$ is a morphism
$(\mc{G}',\mc{F}') \rightarrow (\mc{G},\mc{F})$.  By
Lemma~\ref{lem-mor}, for each $r=0,\dots,g-1$, there exists a
quasi-compact, separated, smooth morphism
$\tau^r(\zeta):\text{GL}(\mc{F})\times_S \text{GL}(\mc{G})\times_S
M^{(r)}(S,\mc{G}', \mc{F}') \rightarrow M^{(r+1)}(S,\mc{G},\mc{F})$
whose image is $U^{(r+1)}$.

\medskip\noindent
Since $\tau^{r-1}(\zeta): \text{GL}(\mc{F}) \times_S \text{GL}(\mc{G})
\times_S M^{(r-1)}(S,\mc{G}',\mc{F}') \rightarrow U^{(r)}$ is smooth
and surjective, to prove that $U^{(r)}$ is smooth over $S$ it suffices
to prove that $\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S
M^{(r-1)}(S,\mc{G}',\mc{F}')$ is smooth over $S$.  By the induction
hypothesis, $M^{(r-1)}(S,\mc{G}',\mc{F}')$ is smooth over $S$.  And
$\text{GL}(\mc{F})$ and $\text{GL}(\mc{G})$ are obviously smooth over
$S$.  Therefore the fiber product is smooth over $S$.  This
proves (i) over $U^{(r)}$.

\medskip\noindent
\textbf{(ii); $U^{(r)}$}:
Similarly, to prove that $U^{(r)} \cap \lt( E^{(r)}_0 \cup \dots \cup
E^{(r)}_{r-1} \rt)$ is a simple normal crossings divisor, it suffices
to prove that the pullback under $\tau^{r-1}(\zeta)$ is a simple
normal crossings divisor.  By (vi) of Lemma~\ref{lem-mor}, this
pullback is of the form $\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G})
\times_S E$ where $E = E^{(r-1)}_0 \cup \dots E^{(r-1)}_{r-2}$ (the
``missing'' divisor is due to the fact that the pullback of
$E^{(r)}_0$ is the empty set).  By the induction hypothesis, $E$ is a
simple normal crossings divisor in $M^{(r-1)}(S,\mc{G}',\mc{F}')$.
Therefore $\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S E$ is
a simple normal crossings divisor in $\text{GL}(\mc{F}) \times_S
\text{GL}(\mc{G}) \times_S M^{(r-1)}(S,\mc{G}',\mc{F}')$.  Hence
$U^{(r)} \cap \lt( E^{(r)}_0 \cup \dots \cup E^{(r)}_{r-1} \rt)$
is a simple normal crossings divisor in $U^{(r)}$.  This proves
(ii) over $U^{(r)}$.

\medskip\noindent
\textbf{(iii); $U^{(r)}$}:
To prove that $U^{(r)}\cap M^{(r)}_r$ is smooth over $S$, it suffices
to prove that the pullback under $\tau^{r-1}(\zeta)$ is smooth over
$S$.  By (v) of Lemma~\ref{lem-mor}, the pullback is
$\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S
M^{(r-1)}_{r-1}(S,\mc{G}', \mc{F}')$.  By the induction hypothesis
$M^{(r-1)}_{r-1}(S,\mc{G}',\mc{F}')$ is smooth over $S$, so also
$\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S
M^{(r-1)}_{r-1}(S,\mc{G}',\mc{F}')$ is smooth over $S$.  It follows
that $U^{(r)} \cap M^{(r)}_r$ is smooth over $S$.  This establishes
(iii) over $U^{(r)}$.

\medskip\noindent 
\textbf{(v); $U^{(r)}$}:
Item (iv) is quite a bit more involved.  By the induction hypothesis,
the maps $(\psi')^{(r)}:(u^{r-1,r})^*\lt(\mc{G}' \rt)^{(r-1)}
\rightarrow \lt(\mc{G}'\rt)^{(r)}$ and the maps $(\phi')^{(r)}: \lt(
\mc{G}' \rt)^{(r)} \rightarrow (u^{0,r})^* \mc{F}'$ on
$M^{(r)}(S,\mc{G}',\mc{F}')$ are all defined and satisfy the
conditions in (iv) and (v).  First 
(v) is proved on $U^{(r)}$.  Observe that if the sequence of maps
$\psi^{(s)},\phi^{(s)}$ exists for $s=0,\dots,r-1$, then there is \emph{at
most one pair} $\psi^{(r)},\phi^{(r)}$ satisfying the hypotheses.
This is because the restriction of $(u^{r-1,r})^*\phi^{(r-1)}$ to
$E^{(r)}_{r-1}$ has rank at least $r-1$.  So the kernel has rank at
least $g+1-r$.  If there exists a pair $\psi^{(r)},\phi^{(r)}$, then
the restriction of $(u^{r-1,r})^*\phi^{(r-1)}$ to
$E^{(r)}_{r-1}$ has constant rank $r-1$ (i.e. the cokernel is locally
free of rank $g+1-r$) and $\psi^{(r)}: (u^{r-1,r})^*\mc{G}^{(r-1)}
\rightarrow \mc{G}^{(r)}$ must be the elementary-transform-up along
$E^{(r)}_{r-1}$ whose kernel equals the kernel of
$(u^{r-1,r})^*\phi^{(r-1)}$.  And then $\phi^{(r)}$ is the unique
morphism through which $(u^{r-1,r})^*\phi^{(r-1)}$ factors.  This
proves (v) (in fact, without restricting over $U^{(r)}$).
Equivariance with respect to $\text{GL}(\mc{F}) \times_S
\text{GL}(\mc{G})$ follows by induction on $r$ and the uniqueness just
mentioned.

\medskip\noindent
\textbf{(iv); $U^{(r)}$}:
Next the existence of $\psi^{(r)},\phi^{(r)}$ is proved when restricted
over $U^{(r)}$.  This is proved by faithfully flat (in fact
smooth) descent, i.e. a descent datum is constructed for the
faithfully flat cover $\tau^{r-1}(\zeta): \text{GL}(\mc{F}) \times_S
\text{GL}(\mc{G}) \times_S M^{(r-1)}(S,\mc{G}',\mc{F}') \rightarrow
U^{(r)}$.  The uniqueness in (iv) and equivariance with respect
to $\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G})$ will give
the cocycle condition.

\medskip\noindent
For each $r=1,\dots,g$ define $\mc{G}^{(r)}_{\text{pre}}$ on
$M^{(r-1)}(S,\mc{G}',\mc{F}')$ to be the direct sum of $\lt( \mc{G}'
\rt)^{(r-1)}$ and $\text{Ker}(p) \otimes_{\OO_S} \OO_{M^{(r-1)}}$.  In
particular, $\mc{G}^{(1)}$ is simply $\mc{G} \otimes_{\OO_S}
\OO_{M^{(0)}}$.  For each $r=2,\dots, g$ define
$\psi^{(r)}_{\text{pre}}: (u^{r-2,r-1})^*\mc{G}^{(r-1)}_{\text{pre}}
\rightarrow \mc{G}^{(r)}_{\text{pre}}$ to be the direct sum of
$(\psi')^{(r-1)}:(u^{r-2,r-1})^* \lt( \mc{G}' \rt)^{(r-2)} \rightarrow
\lt( \mc{G}' \rt)^{(r-1)}$ and the identity map on $\text{Ker}(p)
\otimes_{\OO_S} \OO_{M^{(r-1)}}$.  For each $r=1,\dots, g$ define
$\phi^{(r)}_{\text{pre}}: \mc{G}^{(r)}_{\text{pre}} \rightarrow
\mc{F}\otimes_{\OO_S} \OO_{M^{(r-1)}}$ to be the sum of the map
\begin{equation}
q\circ
(\phi')^{(r-1)}: \lt( \mc{G}' \rt)^{(r-1)} \rightarrow \mc{F}'
\otimes_{\OO_S} \OO_{M^{(r-1)}} \rightarrow \mc{F} \otimes_{\OO_S}
\OO_{M^{(r-1)}}
\end{equation}
with the map $T: \text{Ker}(p)\otimes_{\OO_S} \OO_{M^{(r-1)}}
\rightarrow \mc{F} \otimes_{\OO_S} \OO_{M^{(r-1)}}$.  

\medskip\noindent
On $\text{GL}(\mc{F})$ there is a universal automorphism
$\alpha:\mc{F} \otimes_{\OO_S} \OO \rightarrow \mc{F} \otimes_{\OO_S}
\OO$ and on $\text{GL}(\mc{F})$ there is a universal automorphism
$\beta:\mc{G} \otimes_{\OO_S} \OO \rightarrow \mc{G} \otimes_{\OO_S}
\OO$.  By slight abuse of notation, also denote by 
$\alpha$ and $\beta$ the pullbacks of these
automorphisms to $\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G})
\times_S M^{(r-1)}(S,\mc{G}', \mc{F}')$.  By definition, the pullback
by $\tau^{r-1}(\zeta)$ of $(u^{0,r})^*\phi$ equals $\alpha \circ
(u^{0,r-1})^*\text{pr}_3^*\phi^{(1)}_{\text{pre}} \circ \beta^{-1}$.
Next the part of the descent datum on $\text{GL}(\mc{F})
\times_S \text{GL}(\mc{G}) \times_S M^{(r-1)}(S,\mc{G}',\mc{F}')$
defining $\mc{G}^{(r)}$, $\psi^{(r)}$ and $\phi^{(r)}$ is specified.  
For each
$r=1,\dots, g$ define $\tau^{r}(\zeta)^* \mc{G}^{(r)}$ to be
$\text{pr}_3^* \mc{G}^{(r)}_{\text{pre}}$.  Define
$\tau^{0}(\zeta)^* \psi^{(1)}$ to be $\beta^{-1}$, where the domain of
$\beta^{-1}$ is identified with $\mc{G}\otimes_{\OO_S} \OO \cong
\tau^{0}(\zeta)^* (u^{0,1})^* \mc{G}^{(0)}$ and where the range of
$\beta^{-1}$ is identified with $\mc{G}\otimes_{\OO_S} \OO \cong
\text{pr}_3^* \mc{G}^{(1)}_{\text{pre}}$.  For $r=2,\dots,g$, 
define $\tau^{r-1}(\zeta)^* \psi^{(r)}$ to be $\text{pr}_3^*
\psi^{(r)}_{\text{pre}}$.  For all $r=1,\dots,g$, define
$\tau^{r-1}(\zeta)^* \phi^{(r)}$ to be $\alpha \circ \text{pr}_3^*
\phi^{(r)}_{\text{pre}}$.

\medskip\noindent
Now to finish specifying the descent data, patching
morphisms on the fiber product of $\tau^{r-1}(\zeta)$ 
with itself are needed.
There are canonical descent data associated to the sheaves $\mc{G}
\otimes_{\OO_S} \OO_{M^{(r)}}$ and $\mc{F} \otimes_{\OO_S}
\OO_{M^{(r)}}$ on $M^{(r)}(S,\mc{G},\mc{F})$.  And, up to unique
isomorphism, there is \emph{at most one way} of extending the descent data
for $\mc{G}^{(r)}$, $\psi^{(r)}$ and $\phi^{(r)}$ so that the descent
data giving $\psi^{(r)}$ and $\phi^{(r)}$ are morphisms from the
descent datum for $\mc{G} \otimes_{\OO_S} \OO_{M^{(r)}}$ to the
descent datum for $\mc{G}^{(r)}$ and from the descent datum for
$\mc{G}^{(r)}$ to the descent datum for $\mc{F} \otimes_{\OO_S}
\OO_{M^{(r)}}$ respectively.  This doesn't prove that such descent
data exists!  Proving existence is an exercise in the compatibilities
of all the sheaves and morphisms defined so far, and is left to the
reader.  The key point, as always, is that on
$M^{(r)}(S,\mc{G},\mc{F})$, on the base-change by $\tau^{r-1}(\zeta)$,
and on the double base-change by $\tau^{r-1}(\zeta)$, the morphism
$\psi^{(r)}$ is, up to unique isomorphism, the elementary-transform-up
determined by the kernel of $(u^{r-1,r})^*\phi^{(r-1)}$ restricted to
$E^{(r)}_{r-1}$.

\medskip\noindent
The upshot is that the sheaves $\mc{G}^{(r)}$ and sheaf maps
$\psi^{(r)}$, $\phi^{(r)}$ exist when restricted over $U^{(r)}$.  
Checking the properties in (iv), i.e. that the cokernel of
$\psi^{(r)}$ is as specified and that the rank of $\phi^{(r)}$ is as
specified, can be done after base-changing by $\tau^{r-1}(\zeta)$.  And
then it follows from the construction of the descent datum, and by the
induction hypothesis applied to $M^{(r-1)}(S,\mc{G}',\mc{F}')$.
Again, the details are left to the interested reader.

\medskip\noindent
\textbf{(vi); $U^{(r)}$}:
Next (vi) is proved when restricted over $U^{(r)}$, or
rather over $U^{(s)}$.
The map $v^{1,0}:U^{(1)} \rightarrow U^{(0)}$ is an isomorphism, so 
it suffices to consider the cases $1 \leq r < s \leq g$.  
To check that two ideal
sheaves are equal, it suffices to check after faithfully flat base-change. 
So it suffices to check after base-change by $\tau^{s-1}(\zeta)$.  
By (iii) of
Lemma~\ref{lem-mor}, the inverse image under $\tau^{s-1}(\zeta)$ of
the inverse image under $u^{r,s}$ equals the inverse image under
$\text{Id} \times \text{Id} \times u^{r-1,s-1}$ of
$\tau^{r-1}(\zeta)$.  By (v) of Lemma~\ref{lem-mor}, the
inverse image under $\tau^{r-1}(\zeta)$ of $\mc{I}^{(r)}_k$ equals the
ideal sheaf of the closed subscheme $\text{GL}(\mc{F}) \times_S
\text{GL}(\mc{G}) \times_S M^{(r-1)}_{k-1}(S,\mc{G}',\mc{F}')$ of
$\text{GL}(\mc{F}) \times_S \text{GL}(\mc{G}) \times_S
M^{(r-1)}(S,\mc{G}',\mc{F}')$.  By the induction hypothesis and (v), 
the inverse image of $\mc{I}^{(r-1)}_{k-1}(S,\mc{G}',\mc{F}')$
under $u^{r-1,s-1}$ is the product of
$\mc{I}^{(s-1)}_{k-1}(S,\mc{G}',\mc{F}')$ with the invertible ideal
sheaf associated to the Cartier divisor $\sum_{j=r-1}^{s-2}
((k-1)+1-j) E^{(s-1)}_j$.  Making the substitution $i=j+1$, the
Cartier divisor is $\sum_{i=r}^{s-1}(k+1-i) E^{(s-1)}_{i-1}$.  Taking
the inverse image of this ideal sheaf under $\text{pr}_3$, and using
(v) and (vi) of Lemma~\ref{lem-mor}, this gives the same
ideal sheaf as the inverse image under $\tau^{s-1}(\zeta)$ of the
ideal sheaf in (vi) above.  This establishes (vi) over
$U^{(r)}$.  This finishes the proof of the proposition ``over $U^{(r)}$''.

\medskip\noindent
\textbf{(i), $M^{(r)}$}:
To finish the induction step, (i)--(vi) have to be proved over 
all of $M^{(r)}(S,\mc{G},\mc{F})$.  This is done
using the morphisms $f^r:\AAA^1 \times U^{(r)} \rightarrow M^{(r)}$
from Lemma~\ref{lem-f}.  By (iv) of Lemma~\ref{lem-f}, the
morphism $f^r$ is smooth and surjective, so to check the target of
$f^r$ is smooth over $S$, it suffices to check the domain of $f^r$ is
smooth over $S$.  As established above, $U^{(r)}$ is smooth over $S$
so that $\AAA^1 \times U^{(r)}$ is smooth over $S$.  Therefore
$M^{(r)}$ is smooth over $S$.  This proves (i) over all of
$M^{(r)}$.

\medskip\noindent
\textbf{(ii), $M^{(r)}$}:
Similarly, to show that $E^{(r)}_0 \cup \dots \cup E^{(r)}_{r-1}$ is a
simple normal crossings divisor, it suffices to prove that the
pullback by $f^{r}$ is a simple normal crossings divisor.  By (vi)
of Lemma~\ref{lem-f}, the preimage of $E^{(r)}_0$ is $\{0\}
\times U^{(r)}$ and the preimage of $E^{(r)}_1 \cup \dots \cup
E^{(r)}_{r-1}$ is $\AAA^1 \times \lt(U^{(r)} \cap \lt( E^{(r)}_1 \cup
\dots \cup E^{(r)}_{r-1} \rt) \rt)$.  As proved above, $U^{(r)}
\cap \lt( E^{(r)}_1 \cup \dots \cup E^{(r)}_{r-1} \rt)$ is a simple
normal crossings divisor in $U^{(r)}$.  So by Lemma~\ref{lem-snc}, the
divisor $(f^r)^{-1} \lt( E^{(r)}_0 \cup \dots \cup E^{(r)}_{r-1} \rt)$
is a simple normal crossings divisor.  This proves (ii)
over all of $M^{(r)}$.

\medskip\noindent
\textbf{(iii), $M^{(r)}$}:
To show that $M^{(r)}_r$ is smooth over $S$, it suffices to show that
the preimage under $f^r$ is smooth over $S$.  By (v) of
Lemma~\ref{lem-f}, the preimage of $M^{(r)}_r$ is $\AAA^1 \times \lt(
U^{(r)} \cap M^{(r)}_r \rt)$.  As proved above, $U^{(r)} \cap
M^{(r)}_r$ is smooth.  So $\AAA^1 \times \lt( U^{(r)} \cap M^{(r)}_r
\rt)$ is smooth, and therefore $M^{(r)}_r$ is smooth.  This
proves (iii) over all of $M^{(r)}$.

\medskip\noindent
\textbf{(iv) and (v); $M^{(r)}$}:
As over $U^{(r)}$, (iv) and (v) are a bit more involved
(although most of the work is already done).  As before, Item $(5)$
is automatic, once the existence of $\mc{G}^{(r)}$,
$\psi^{(r)}$ and $\phi^{(r)}$ satisfying (iv)
is proved.
Existence is proved by faithfully flat descent with respect to the
faithfully flat (in fact smooth) morphism $f^r:\AAA^1 \times U^{(r)}
\rightarrow M^{(r)}$.  Next the part of the descent data on
$\AAA^1 \times U^{(r)}$ defining $\mc{G}^{(r)}$, $\psi^{(r)}$ and
$\phi^{(r)}$ is specified.  For each $r=1,\dots, g$, define $(f^r)^*
\mc{G}^{(r)}$ to be $\text{pr}_2^* (\mc{G}^{(r)}|_{U^{(r)}})\otimes
\OO_{\AAA^1 \times U^{(r)}}(\{ 0 \} \times U^{(r)})$ (of course this
definition looks circular, but recall that 
$\mc{G}^{(r)}|_{U^{(r)}}$, $\psi^{(r)}|_{U^{(r)}}$ and
$\phi^{(r)}|_{U^{(r)}}$ were already constructed above).  
Let $t$ be the coordinate on $\AAA^1$
considered as a global section of the invertible sheaf $\OO_{\AAA^1
\times U^{(r)}}(\{ 0 \} \times U^{(r)})$ whose vanishing locus is
precisely $\{ 0 \} \times U^{(r)}$ (this conflicts with the
usual terminology that calls this section ``$1$'').  The point is
that there is a canonical everywhere nonzero global section of this
invertible sheaf (which in the usual terminology is denoted by
``$\frac{1}{t}$'', but here is denoted by ``$1$''), and with respect
to this trivialization the regular function $t$ corresponds to a
section whose vanishing locus is $\{ 0 \} \times U^{(r)}$.

\medskip\noindent 
Define $(f^1)^*(\psi^{(1)})$ to be the map
\begin{equation}
\text{Id} \otimes t:
\mc{G} \otimes_{\OO_S} \OO_{\AAA^1 \times U^{(1)}} \rightarrow \mc{G}
\otimes_{\OO_S} \OO_{\AAA^1 \times U^{(1)}} ( \{0\} \times U^{(1)} ).
\end{equation}
Of course the domain of this map is identified with the pullback by
$f^1$ of $\mc{G}^{(0)} = \mc{G} \otimes_{\OO_S} \OO_{M^{(1)}}$ and 
the target is identified with $(f^1)^* \mc{G}^{(1)}$ defined above.  For
$r=2,\dots, g$, define $(f^r)^*(\psi^{(r)})$ to be the map
$\text{pr}_2^*(\psi^{(r)}|_{U^{(r)}})\otimes \text{Id}$.  Define
$(f^1)^*(\phi^{(1)})$ to be the composition of the canonical
isomorphism $\text{Id} \otimes 1: \text{pr}_2^*\mc{G}^{(r)}\otimes
\OO_{\AAA^1 \times U^{(r)}}(\{ 0 \} \times U^{(r)}) \rightarrow
\text{pr}_2^* \mc{G}^{(r)}$ with
$\text{pr}_2^*(\phi^{(r)}|_{U^{(r)}})$.  Observe that $(f^1)^*
\phi^{(1)}$ is the same map constructed in Lemma~\ref{lem-pref}.

\medskip\noindent
Next it is proved these definitions of $(f^r)^*\psi^{(r)}$ and
$(f^r)^*\phi^{(r)}$ have the properties from (iv).  For $r=1$,
this is precisely Lemma~\ref{lem-pref}.  Suppose that $r \geq 2$.  As
established above, $(\phi^{(r)}|_{U^{(r)}}) \circ
(\psi^{(r)}|_{U^{(r)}})$ equals
$(v^{r-1,r})^*(\phi^{(r-1)}|_{U^{(r-1)}})$.  Pulling back by
$\text{pr}_2^*$ and using (iii) from Lemma~\ref{lem-f}, 
$(f^r)^*\phi^{(r)}\circ (f^r)^* \psi^{(r)}$ equals
$(\text{Id} \times v^{r-1,r})^*(f^{r-1})^*\phi^{(r-1)} = (f^r)^*
(u^{r-1,r})^* \phi^{(r-1)}$.  This is the first necessary property.
As established above, the cokernel of $\psi^{(r)}|_{U^{(r)}}$ is the
push forward of a locally free sheaf of rank $g+1-r$ from the divisor
$U^{(r)}\cap E^{(r)}_{r-1}$ in $U^{(r)}$.  Therefore the cokernel of
$(f^r)^* \psi^{(r)}$, i.e. the cokernel of $\text{pr}_2^*
(\psi^{(r)}|_{U^{(r)}})$, is the push forward of a locally free sheaf
of rank $g+1-r$ from the divisor $\AAA^1 \times (U^{(r)} \cap
E^{(r)}_{r-1})$, i.e. the divisor $(f^r)^{-1}(E^{(r)}_{r-1})$.  This
is the second necessary property.  And as established above,
$\phi^{(r)}|_{U^{(r)}}$ has rank at least $r$ at all geometric points.
Therefore $(f^r)^*\phi^{(r)}$,
i.e. $\text{pr}_2^*(\phi^{(r)}|_{U^{(r)}})$, has rank at least $r$ at
all geometric points.  This is the last necessary property.

\medskip\noindent
To finish the proof of (iv), the rest of the descent data must be
specified, i.e.
the patching morphisms on the fiber product of $f^r$
with itself are needed.
As over $U^{(r)}$,
there is at most one way of completing the descent data so that
$\psi^{(r)}$ and $\phi^{(r)}$ give morphisms with the given descent
data for $\mc{G} \otimes_{\OO_S} \OO_{M^{(r)}}$ and $\mc{F}
\otimes_{\OO_S} \OO_{M^{(r)}}$.  Proving that one can complete the
descent data is an exercise left to the reader.  The
key point (as always!) is that $\psi^{(r)}$ is the
elementary-transform-up determined by the kernel of
$(u^{r-1,r})^*\phi^{(r-1)}$ restricted 
to $E^{(r)}_{r-1}$.  This proves (iv) and (v)
on all of $M^{(r)}$.

\medskip\noindent
\textbf{(vi); $M^{(r)}$}:
First suppose that $r=0$.  By (iv) 
of Lemma~\ref{lem-pref}, $(u^{0,1})^{-1}(\mc{I}^{(0)}_k)$ equals
$\mc{I}^{(1)}_k \cdot \OO_{M^{(1)}}(-(k+1)E^{(1)}_0)$.  By (v)
of Lemma~\ref{lem-f}, $(f^1)^{-1}(\mc{I}^{(1)}_k)$ is the ideal sheaf
$\text{pr}_2^{-1}(\mc{I}^{(1)}_k|_{U^{(1)}})$.  By (vi) of
Lemma~\ref{lem-f}, $(f^1)^{-1}\OO_{M^{(1)}}(-E^{(1)}_0)$ is the ideal
sheaf of $\{0\} \times U^{(1)}$.  The process of forming the preimage
of an ideal sheaf by $\text{pr}_2$ and then forming the preimage of
that ideal sheaf by $\text{Id} \times v^{1,s}$ is the same as the
process of first forming the preimage of the ideal sheaf by $v^{1,s}$
and then forming the preimage by $\text{pr}_2$.  As established above,
$(v^{1,s})^{-1}(\mc{I}^{(1)}_k|_{U^{(1)}})$ equals the restriction to
$U^{(s)}$ of the ideal sheaf
\begin{equation}
\mc{I}^{(s)}_k \cdot \OO_{U^{(s)}} (- ( \sum_{i=1}^{s-1} (k+1-i)
E^{(s)}_i ) ).
\end{equation}
And the inverse image under $\text{Id} \times v^{1,s}$ of the ideal sheaf of
$\{0\} \times U^{(1)}$ is the ideal sheaf of $\{ 0 \} \times
U^{(s)}$.  Putting the pieces together, the inverse image under $\text{Id}
\times v^{1,s}$ of the inverse image under $f^1$ of the inverse image
under $u^{0,1}$ of $\mc{I}^{(0)}_k$ equals the ideal sheaf
\begin{equation}
\text{pr}_2^{-1}(\mc{I}^{(s)}_k|_{U^{(s)}}) \cdot \OO_{\AAA^1 \times
  U^{(s)}} \lt( -(k+1)\{0\} \times U^{(s)} \rt) \otimes
\OO_{\AAA^1 \times U^{(s)}}
  \lt( - \lt( \sum_{i=1}^{s-1}(k+1-i) \AAA^1 \times \lt(
  U^{(s)} \cap E^{(s)}_i \rt) \rt) \rt).
\end{equation}
By (v) and (vi) of Lemma~\ref{lem-f}, this is precisely
the inverse image under $f^s$ of the ideal sheaf from (vi).  
Using (iii) of Lemma~\ref{lem-f} one
last time, and using that one can check equality of ideal sheaves
after faithfully flat base-change, (vi) holds
when $r=1$.

\medskip\noindent
Checking (vi) for $r > 1$ is even easier and follows by the
same sort of argument as above; the details are left to the 
reader.  This proves that (vi) holds over all of $M^{(r)}$,
and thus finishes the proof that the proposition holds over all of $M^{(r)}$.
The proposition is proved by induction on the rank of $\mc{G}$.
\end{proof}

\medskip\noindent
\subsection{Computation of the log discrepancies} \label{subsec-comp}
In this section the log resolution from the last section is applied to 
compute the log
discrepancies of the pair $(M^{(0)},M^{(0)}_k)$.  In the application,
it is also necessary
to compute the log discrepancies of the pair $(M^{(0)},q\cdot
M^{(0)}_k)$.  Then, combined with the results of
Section~\ref{sec-discrep}, these computations are used to find the log
discrepancies of some projective cones.

\medskip\noindent
\begin{lem} \label{lem-mults} 
Let $(S,\mc{G},\mc{F})$ be a datum with $\text{rank}(\mc{G}) = g$ and
$\text{rank}(\mc{F}) = f$.  For each $0 \leq r < s \leq g$, the
relative canonical divisor of $u^{r,s}:M^{(s)} \rightarrow M^{(r)}$
equals
\begin{equation}
K_{M^{(s)}} - (u^{r,s})^*K_{M^{(r)}} = \sum_{i=r}^{s-1} \lt(
(f-i)(g-i) -1 \rt) E^{(s)}_{i}.
\end{equation}
For $r< s \leq k < g$ and each positive integer $q$, 
the inverse image under $u^{r,s}$ of the ideal sheaf
$\lt( \mc{I}^{(r)}_k \rt)^q$ equals
\begin{equation}
\lt( \mc{I}^{(s)}_k \rt)^q \cdot \OO_{M^{(s)}} \lt( - q\lt(
\sum_{i=r}^{s-1} (k+1-i) E^{(s)}_i \rt) \rt).
\end{equation}
and the associated cycle is
\begin{equation}
q \lt[ M^{(s)}_{k} \rt] + \sum_{i=r}^{s-1} q(k+1-i)E^{(s)}_{i}
\end{equation}
Finally, for $k < s\leq g$ and each positive integer $q$, the inverse
image under $u^{r,s}$ of the ideal sheaf $\lt( \mc{I}^{(r)}_k \rt)^q$
is an invertible ideal sheaf defining the Cartier divisor
\begin{equation}
\sum_{i=r}^{k} q(k+1-i) E^{(s)}_{i}
\end{equation}
\end{lem}

\medskip\noindent
\begin{proof}
By Proposition~\ref{prop-smooth}, $M^{(r)}_r \subset
M^{(r)}$ is a regular embedding.  And by ~\cite[Prop., p. 67]{ACGH},
the codimension of $M^{(r)}_r$, which equals the codimension of
$M^{(0)}_r$ in $M^{(0)}$, equals $(f-r)(g-r)$.  Therefore the relative
canonical divisor of the blowing up $u^{r,r+1}:M^{(r+1)} \rightarrow
M^{(r)}$ is $(f-r)(g-r)E^{(r+1)}_r$.  The first formula follows since
the relative canonical divisor of a composition of birational
morphisms is the sum of the relative canonical divisors of the
separate morphisms.

\medskip\noindent
The second formula follows from (vi) of
Proposition~\ref{prop-smooth}.  The final formula follows from the
second formula and that fact that
$(u^{r,r-1})^{-1}\mc{I}^{(r-1)}_{r-1}$ equals the invertible ideal
sheaf $\OO_{M^{(r)}}(-E^{(r)}_{r-1})$.
\end{proof}

\medskip\noindent
\begin{cor} \label{cor-mults} 
Let $(S,\mc{G},\mc{F})$ be a datum with $\text{rank}(\mc{G}) = g$ and
$\text{rank}(\mc{F}) = f$.  Suppose that $S$ is smooth, that $g\geq
1$, that $f \geq 2$ and that $0\leq k \leq g-1$.  Consider the pair
$(M^{(r)}(S,\mc{G},\mc{F}), q\cdot M^{(r)}_{k}(S,\mc{G},\mc{F}))$
where $q \geq 0$.  For $i=r,\dots,k$, the log discrepancies are
\begin{equation}
a(E^{(g)}_i;M^{(r)},q\cdot M^{(r)}_{k}) = (f-i)(g-i) - q(k+1-i).
\end{equation}
Define $a = \text{min}\{ (f-i)(g-i) - q(k+1-i)| i=r,\dots,k \}$.  Then
$(M^{(r)}, q\cdot M^{(r)}_{k})$ is log canonical iff $a \geq 0$, in
which case the minimal log discrepancy, $\text{mld}(M^{(r)};M^{(r)},
q\cdot M^{(r)}_{k})$, equals $\text{min}(1,a)$.  
In particular, if $q\leq f-g+1$,
then the pair $(M^{(r)},q\cdot M^{(r)}_{g-1})$ is log canonical and
the minimal log discrepancy equals $\text{min}(1,f-g+1-q)$.
\end{cor}

\medskip\noindent
\begin{proof}
The corollary follows from the computations in Lemma~\ref{lem-mults}
and the definition of the log discrepancies and minimal log
discrepancies of a pair (c.f. ~\cite[Defn. 1.1]{EMY}).
\end{proof}

\medskip\noindent
Let $(S,\mc{G},\mc{F})$ be a datum with $\text{rank}(\mc{G}) = g$ and
$\text{rank}(\mc{F}) = f$.  Let $\mc{A}$ and $\mc{A}'$ be locally free
sheaves on $S$ with $\text{rank}(\mc{A}) = a$ and
$\text{rank}(\mc{A}') =a'$ with $a > 0$.  Define $\mc{E}$ to be the
cokernel of the following sheaf map on $M^{(0)}$,
\begin{equation}
\text{Id}\otimes \phi: \mc{A} \otimes_{\OO_S} \mc{G} \otimes_{\OO_S}
\OO_{M^{(0)}} \rightarrow \mc{A} \otimes_{\OO_S} \mc{F}
\otimes_{\OO_S} \OO_{M^{(0)}} \oplus \mc{A}'\otimes_{\OO_S} \OO_{M^{(0)}}.
\end{equation}
The sheaf map is zero on the summand $\mc{A}' \otimes_{\OO_S}
\OO_{M^{(0)}}$, so this locally free sheaf will be a direct summand of
$\mc{E}$. 
Denote by the pair $(\pi:C\rightarrow M^{(0)}, \alpha:\pi^* \mc{E}
\rightarrow \mc{Q} )$ the relative Grassmannian cone parametrizing
rank $r$ locally free quotients of $\mc{E}$.  

\medskip\noindent
\begin{prop} \label{prop-cone} 
If $a\cdot r \leq f-g$ and if $S$ is smooth and geometrically connected,
then $C$ has pure dimension equal to the expected dimension $d =
\text{dim}(C) = \text{dim}(S) + f\cdot g + r\lt( (a(f-g)+a') - r\rt)$
and $C$ is a normal, integral, local complete intersection scheme
that has, at worst, canonical singularities.
\end{prop}

\medskip\noindent
\begin{proof}
Denote $g' = a\cdot g = \text{rank}(\mc{A} \otimes_{\OO_S} \mc{G})$.
For each $k=0,\dots, g-1$ and $l=0,\dots,a-1$, $B_{a\cdot
k + l} = B_{a \cdot k} = M^{(0)}_k$.  And this has codimension
$(f-k)(g-k)=(f-g)(g-k) + (g-k)^2$.  Also $r(g'-(a\cdot k + l))+1 =
a\cdot r(g-k)+1 - rl$.  By assumption, $f-g \geq a\cdot r$, and for $k
\leq g-1$, $(g-k)^2 \geq 1$.  Therefore $(f-g)(g-k) + (g-k)^2
\geq a\cdot r(g-k) +1 -rl$ for all $k=0,\dots, g-1$ and
$l=0,\dots,a-1$.  Therefore, by Lemma~\ref{lem-codims}, 
$C$ is irreducible of the expected dimension.  And by the proof of
Lemma~\ref{lem-discrep}, $C$ is a local complete intersection scheme.
In particular it is Cohen-Macaulay.  Moreover, $\pi:C \rightarrow
M^{(0)}$ is smooth over $M^{(0)}-M^{(0)}_{g-1}$.  So $C$ is
generically reduced.  Since $C$ is Cohen-Macaulay, it follows that $C$
is everywhere reduced.  A reduced Cohen-Macauly scheme satisfies
Serre's condition $S2$ for normality.  Thus to prove that $C$ is
normal, it suffices to prove that $C$ is regular in codimension $1$.

\medskip\noindent
If $a\cdot r < f-g$, then the same parameter count as above shows that
for all $k=0,\dots, g-1$ and $l=0,\dots, a-1$ we have that $(f-k)(g-k)
\geq r(g'-(a\cdot k + l)) + 2$.  By Lemma~\ref{lem-codims} it follows
that $C$ is regular in codimension $1$ so that $C$ is normal.
Therefore assume that $a\cdot r = f-g$.  Of course $C$ is regular on
the dense open subset $\pi^{-1}(M^{(0)} - M^{(0)}_{g-1})$.  The only
codimension one point not contained in this locus is the generic point
of $\pi^{-1}(M^{(0)}_{g-1} - M^{(0)}_{g-2})$.

\medskip\noindent
As in the proof of Lemma~\ref{lem-codims}, denote by $(\rho:C'
\rightarrow M^{(0)}, \beta: \rho^*\mc{F} \rightarrow \mc{Q}')$ the
Grassmannian bundle parametrizing rank $r$ locally free quotients of
$\mc{A} \otimes_{\OO_S} \mc{F} \otimes_{\OO_S} \OO_{M^{(0)}} \oplus
\mc{A}'\otimes_{\OO_S} \OO_{M^{(0)}}$.  There is a natural closed
immersion $h:C \rightarrow C'$ compatible with projection to
$M^{(0)}$.

\medskip\noindent
Observe that $C' = P \times_S M^{(0)}$ where $\sigma:P \rightarrow S$
is the Grassmannian bundle parametrizing rank $r$ locally free
quotients of $\mc{A} \otimes_{\OO_S} \mc{F} \oplus \mc{A}'$.
Therefore there is a projection $\text{pr}_2:C \rightarrow P$
compatible with projection to $S$.  The question is local, so it
suffices to 
base-change to an open subset of $S$ over which $\mc{A}$ is trivial.
Choose an ordered basis for $\mc{A}$ so that $\mc{A} \otimes_{\OO_S}
\mc{F}$ is just $\mc{F}^{\oplus a}$.  Let $W \subset P$ denote the
dense open set over which the sheaf map $(\pi')^*(\mc{A}
\otimes_{\OO_S} \mc{F}) \rightarrow \mc{Q}'$ is surjective.  On $W$
there is a smooth, surjective morphism to the Grassmannian $P'$
parametrizing rank $r$ locally free quotients of $\mc{A}
\otimes_{\OO_S} \mc{F}$.  Any rank $r$ quotient space of $\mc{A}
\otimes_{\OO_S} \mc{F}$, i.e. of $\mc{F}^{\oplus a}$, is represented as
the image of a matrix $\mc{F}^{\oplus a} \rightarrow \OO^{\oplus r}$
of the form
\begin{equation}
M = \lt[ \begin{array}{cccc}
v_{1,1} & v_{1,2} & \dots & v_{1,a} \\
\vdots & \vdots  & \ddots & \vdots \\
v_{r,1} & v_{r,2} & \dots & v_{r,a}
\end{array} \rt]
\end{equation}
where the $v_{i,j}$ are sections of $\mc{F}^\vee$.  This matrix
determines a point $x'$ in $P'$.  Let $x\in W$ be any point mapping to
$x'$.  Thinking of the fiber of $\text{pr}_2:C \rightarrow P'$ over
$x$ as a subscheme of $M^{(0)}$, it is just the subscheme of
matrices $L :\mc{G} \rightarrow \mc{F}$ such that $M\circ L^{\oplus
a}$ is the zero matrix, i.e. it is the set of matrices $L$ such that
the kernel of the transpose matrix $L^\dagger$ contains the subspace
\begin{equation}
K = \text{span}\{v_{i,j} | 1\leq i \leq r, 1\leq j \leq a \}.
\end{equation}
This is the space of matrices $L^\dagger$ from $\mc{F}^\vee / K$ to
$\mc{G}^\vee$.  Let $V' \subset P'$ be the dense open subset
parametrizing quotients where $\text{dim}(K) = a\cdot r$ and let $V
\subset W$ be the preimage of $V$.  Over $V$,
$\text{pr}_2:C \rightarrow P$ is a vector bundle of rank $(f-a\cdot
r)g$.  In particular, the preimage is a nonempty, smooth scheme,
i.e. $\text{pr}_2^{-1}(V)$ is contained in the smooth locus
$C_{\text{smooth}}$.  But of course, the map $L^\dagger$ may still
have any rank between $0$ and $g$ (recall that $f-a\cdot r \geq g$, so
that the dimension of $\mc{F}^\vee /K$ is greater than the dimension
of $\mc{G}^\vee$).  Therefore this open set intersects the preimage of
every strata $M^{(0)}_k - M^{(0)}_{k-1}$.  Combined with an obvious
homogeneity argument, the open set intersects every fiber of $\pi$.
So the smooth locus $C_{\text{smooth}}$ intersects every fiber of
$\pi$ and $C$ is regular in codimension $1$ points.
Therefore, in every case, $C$ is normal.

\medskip\noindent
The $(a\cdot (f-g)+a')^{\text{th}}$ Fitting ideal $\mc{J}$ of
$\text{Id} \otimes \phi$, i.e. the ideal generated by the maximal
minors of the matrix of $\text{Id}\otimes \phi$, is easily seen to be
$\mc{I}^a$ where $\mc{I}$ is the $(f-g)^{\text{th}}$ Fitting ideal of
$\mc{I}$, i.e. the ideal sheaf of $M^{(0)}_{g-1}$.  By
Proposition~\ref{prop-relcan}, the cone $(C,\emptyset)$ is canonical
iff the pair $(M^{(0)}, a\cdot r \cdot M^{(0)}_{g-1} )$ is canonical.
And by Corollary~\ref{cor-mults}, $(M^{(0)},a\cdot r \cdot
M^{(0)}_{g-1})$ is canonical.  Therefore $(C,\emptyset)$ is canonical,
which finishes the proof.
\end{proof}

\begin{rmk} \label{rmk-cone} 
When $a\cdot r > f-g$, the cone $C$ has more than one irreducible
component.  It would be interesting to determine the minimal log
discrepancies of the different irreducible components of $C$, in
particular of the unique irreducible component which dominates
$M^{(0)}$.  The first case is when $a\cdot r=f-g+1$.  In this case it
follows from the proof above that the second irreducible component is
the closure of the preimage of $M^{(0)}_{g-1} - M^{(0)}_{g-2}$, and
that the restriction of $\text{pr}_2$ to this irreducible component is
birationally a vector bundle of rank $(f-a\cdot r)g$ over $P$.  It may
be possible to use this structure to compute the minimal log
discrepancies of the two irreducible components.
\end{rmk}

\section{Adjunction for $(B,B_{g-1})$} \label{sec-spec} 
\medskip\noindent
Let $K$ be a field, not necessarily algebraically closed nor of
characteristic zero.  In this section all schemes are $K$-schemes.
$K$-schemes.  We use the results of the last section.  The interested
reader will see how to prove analogous results over an
arbitrary base scheme.

\medskip\noindent
In Section~\ref{sec-discrep} the log discrepancies of a
Grassmannian cone $C \rightarrow B$ were related to the log
discrepancies of the pair $(B,B_{g-1})$.  In this section
the following question is considered:  Let $S\subset B$ be a Cartier
divisor.  If $(S,S_{g-1})$ is log canonical (resp. Kawamata log
terminal, canonical), is $(B,B_{g-1})$ log canonical (resp. Kawamata
log terminal, canonical) on a Zariski open set containing $S$?
This question is a version of ``inversion of adjunction''.  
Combining theorems by Koll\'{a}r and
Shokurov about inversion of adjunction with some new constructions
gives some answers to this question.

\begin{hyp} \label{hyp-spec}
Throughout this section 
$\mc{E}$ is torsion-free of rank $e=f-g > 0$.
\end{hyp}

\medskip\noindent
Let the pair $(\rho:C' \rightarrow B, \beta:\rho^* \mc{F} \rightarrow
\mc{Q}')$ denote the Grassmannian bundle of rank $g$ locally free
quotients of $\mc{F}$.  The first construction equates the log
discrepancies of the pair $(B,B_{g-1})$ with the log discrepancies of
a pair $(C',\mc{D}_{\phi})$ for a Cartier divisor $\mc{D}_{\phi}
\subset C'$.  Morally, the construction is a version of the following
well-known principle (pointed-out to me by Koll\'ar).

\begin{prin} \label{prin-1}
A pair $(B,Z)$ is log canonical (resp. Kawamata log terminal,
canonical) if for a general hypersurface $H\subset B$ containing $Z$,
the pair $(B,H)$ is log canonical (resp. Kawamata log terminal,
canonical).  
\end{prin}

\medskip\noindent
Indeed, if we locally trivialize $\mc{F}$ so that $C'
\cong B \times \text{Grass}(g,f)$, then the fibers of $\mc{D}_{\phi}
\rightarrow \text{Grass}(g,f)$ considered as subvarieties of $B$ are
hypersurfaces containing $B_{g-1}$.  Therefore one expects that the
general fiber of the pair $(C',\mc{D}_{\phi})$ is log canonical,
etc. iff $(B,B_{g-1})$ is log canonical, etc.  In this case, it is
even true that $(C',\mc{D}_{\phi})$ is log canonical, etc. iff
$(B,B_{g-1})$ is log canonical, etc.

\medskip\noindent
\begin{notat} \label{notat-Dphi} 
Denote by $\delta:\rho^* \mc{G} \rightarrow \mc{Q}'$ the composition
\begin{equation}
\delta: \rho^* \mc{G} \xrightarrow{\rho^* \phi} \rho^* \mc{F}
\xrightarrow{\beta} \mc{Q}'.
\end{equation}
Both $\rho^* \mc{G}$ and $\mc{Q}'$ have rank $g$, so $\delta$
induces a well-defined morphism of invertible sheaves
$\text{det}(\delta):\rho^*\text{det}(\mc{G}) \rightarrow
\OO_{C'}(1)$.  Denote by $\mc{D}_{\phi} \subset C'$ the zero
scheme of this morphism.
\end{notat}

\begin{lem} \label{lem-D} 
The Cartier divisor $\mc{D}_{\phi}$ is irreducible and generically
reduced, and the projection morphism $\pi:\mc{D}_{\phi} \rightarrow B$
admits a dualizing complex of the form
\begin{equation}
\omega_{\mc{D}_{\phi}/B} = \rho^*\lt( \text{det}(\mc{F})^{\otimes(r-1)}
  \otimes_{\OO_B} \text{det}(\mc{E}) \rt)\otimes_{\OO_{C'}}
  \OO_{C'}(-(f-1))|_{\mc{D}_{\phi}} [r(f-r) - 1].
\end{equation}
If $B$ is Cohen-Macaulay (resp. Gorenstein) then also $\mc{D}_{\phi}$
is Cohen-Macaulay (resp. Gorenstein).  If $B$ is Cohen-Macaulay and
$\text{codim}_B(B_{g-1}) \geq 3$, then $\mc{D}_{\phi}$ is normal.
\end{lem}

\medskip\noindent
\begin{proof} 
The proof is the same sort of argument as in the proofs of
Lemma~\ref{lem-codims} and Lemma~\ref{lem-discrep}.  The details are
left to the reader.  There is one extra detail in the proof of the
last claim: over $B-B_{g-1}$, the divisor $\mc{D}_{\phi}$ is not
necessarily smooth.  However the singular locus of $\mc{D}_{\phi}$ is
the locus where $\delta$ has rank at most $g-2$, and this has
codimension at least $4$ in $\rho^{-1}(B - B_{g-1})$.  So the singular
locus of $\mc{D}_{\phi} \cap \rho^{-1}(B-B_{g-1})$ has codimension at
least $3$ in $\mc{D}_{\phi}$.  If $\text{codim}_B(B_{g-1}) \geq 3$,
then the codimension of $\mc{D}_{\phi} \cap \rho^{-1}(B_{g-1})$ in
$C'$, i.e. the codimension of $\rho^{-1}(B_{g-1})$ in $C'$, equals
$\text{codim}_B(B_{g-1}) \geq 3$.  Therefore the codimension of
$\rho^{-1}(B_{g-1})$ in $\mc{D}_{\phi}$ is $\text{codim}_B(B_{g-1}) -
1 \geq 2$.  So the singular locus of $\mc{D}_{\phi}$ has codimension
at least $2$ in $\mc{D}_{\phi}$, and, by Serre's criterion, 
$\mc{D}_{\phi}$ is normal.
\end{proof}

\begin{rmk} \label{rmk-D} 
The last condition for $\mc{D}_{\phi}$ to be normal is not a necessary
condition.  It seems certain that $\mc{D}_{\phi}$ is normal provided
that for every codimension $2$ point $\eta$ of $B$ contained in
$B_{g-1}$, $B$ is regular at $\eta$ and $\mc{I}_{g-1}\cdot
\OO_{B,\eta}$ equals the maximal ideal $\mathfrak{p}_\eta$.  In fact
it seems likely that $\mc{D}_{\phi}$ is normal provided that for every
codimension $2$ point $\eta$ of $B$ contained in $B_{g-1}$, $B$ is
regular at $\eta$ and $\mc{I}_{g-1} \cdot \OO_{B,\eta}/
\mathfrak{p}_{\eta}^2$ inside of
$\mathfrak{p}_\eta/\mathfrak{p}_\eta^2$ has dimension at least $1$ as
a $\kappa(\eta)$-vector space.  For example, let
$B=\AAA^2$ and $\phi:\OO_{\AAA^2} \rightarrow \OO_{\AAA^2}^{\oplus 2}$
be the map with matrix $(x,y^m)^\dagger$ for $m\geq 1$.  Then
$\mc{D}_{\phi}$ is normal.
\end{rmk}

\medskip\noindent 
Let $u:\widetilde{B} \rightarrow B$ be a resolution of $\mc{E}$, and
let $\widetilde{\mc{E}}$ and $\widetilde{\phi}: \widetilde{\mc{G}}
\rightarrow u^* \mc{F}$ be as in Notation~\ref{notat-res}.  Inside of
the fiber product $\widetilde{B} \times_B C'$, there is the Cartier
divisor $\mc{D}_{\widetilde{\phi}}$.  The projection morphism
$\text{pr}_2:\widetilde{B} \times_B C' \rightarrow C'$ maps
$\mc{D}_{\widetilde{\phi}}$ onto $\mc{D}_{\phi}$.  Denote by
$w:\mc{D}_{\widetilde{\phi}} \rightarrow \mc{D}_{\phi}$ the induced
morphism.

\begin{lem} \label{lem-relcan2} 
The support of the 
inverse image Cartier divisor $\text{pr}_2^*\mc{D}_\phi$ 
is contained in the divisor $\mc{D}_{\widetilde{\phi}} \cup
\text{pr}_1^{-1}(E_1 \cup \dots E_k)$.  Moreover,
\begin{equation}
K_{\widetilde{B} \times_B C'} + \mc{D}_{\widetilde{\phi}} =
\text{pr}_2^*\lt( K_{C'} + \mc{D}_{\phi} \rt) +   
\sum_{i=1}^k \lt(a(E_i;B,B_{g-1}) -1\rt) \text{pr}_1^* E_i.
\end{equation}
\end{lem}

\begin{proof}
Over $\widetilde{B} - (E_1 \cup \dots \cup E_k)$, it is clear that
$\text{pr}_2$ is an isomorphism.  Therefore,
\begin{equation}
K_{\widetilde{B} \times_B C'} + \mc{D}_{\widetilde{\phi}} =
\text{pr}_2^*\lt( K_{C'} + \mc{D}_{\phi} \rt) +   
\sum_{i=1}^k \lt(a_i -1\rt) \text{pr}_1^* E_i
\end{equation}
for some sequence of rational numbers $a_1,\dots,a_k$.  To compute the
integers $a_i$, first restrict to $\mc{D}_{\widetilde{\phi}}$.  By
adjunction, the restriction of $K_{\widetilde{B} \times_B C'} +
\mc{D}_{\widetilde{\phi}}$ to $\mc{D}_{\widetilde{\phi}}$ is just
$K_{\mc{D}_{\widetilde{\phi}}}$.  And the restriction of
$\text{pr}_2^*(K_{C'} + \mc{D}_{\phi})$ equals $w^*K_{\mc{D}_{\phi}}$
(which is \emph{defined} to be
the sum of $K_B$ and $C_1(\omega_{\mc{D}_{\phi}/B})$
if $\mc{D}_{\phi}$ is not normal).  Applying
Lemma~\ref{lem-D} to both $\mc{D}_{\phi}$ and
$\mc{D}_{\widetilde{\phi}}$, 
\begin{equation}
\begin{array}{ll}
K_{\mc{D}_{\widetilde{\phi}}} - w^*K_{\mc{D}_{\phi}} = \\
\widetilde{\pi}^* \lt[ K_{\widetilde{B}} - u^*K_B +
C_1( \text{det}(\widetilde{\mc{E}}) ) - C_1( \text{det}(\mc{E}) ) \rt].
\end{array} 
\end{equation}
As proved in Lemma~\ref{lem-relcan}, the divisor on the right is just
$\sum_{i=1}^k \lt( a(E_i;B,B_{g-1}) - 1 \rt)\widetilde{\pi}^*E_i$.  So
the restriction to $\mc{D}_{\widetilde{\phi}}$ of
$\sum_{i=1}^k \lt(a_i - 1\rt) \text{pr}_1^* E_i$ equals the restriction of
$\sum_{i=1}^k \lt( a(E_i;B,B_{g-1}) - 1 \rt) \text{pr}_1^* E_i$.

\medskip\noindent
To finish the argument, it must be proved that the pullback map
$\text{Pic}(\widetilde{B}) \rightarrow
\text{Pic}(\mc{D}_{\widetilde{\phi}})$ is injective.  There are
two possible cases:
$f-g \leq g$ or $f-g > g$.

\medskip\noindent
Suppose first that $f-g \leq g$.  Form the Grassmannian bundle
$(\sigma:C'' \rightarrow \widetilde{B}, \epsilon:\sigma^*
\widetilde{\mc{G}} \rightarrow \mc{Q}'')$ parametrizing rank $g-(f-g)$
locally free quotients of $\widetilde{\mc{G}}$.  The coproduct of
$\epsilon:\sigma^* \widetilde{\mc{G}} \rightarrow \mc{Q}''$ and
$\sigma^*\widetilde{\phi}: \sigma^* \widetilde{\mc{G}} \rightarrow
\sigma^* u^*\mc{F}$ gives a surjective morphism of sheaves $\sigma^*
u^*\mc{F} \rightarrow \mc{R}$ where $\mc{R}$ is locally free of rank
$\text{rank}(\mc{F}) - \text{rank}(\widetilde{\mc{G}}) +
\text{rank}(\mc{Q}'') = g$.  There is an induced morphism from $C''$
to $\widetilde{B} \times_B C$ compatible with the projection
to $\widetilde{B}$.  And the image is contained in
$\mc{D}_{\widetilde{\phi}}$.  Since $\sigma:C'' \rightarrow
\widetilde{B}$ is a Grassmannian bundle, the pullback map on Picard
groups is injective.  And this map factors through the pullback map on
Picard groups from $\widetilde{B}$ to $\mc{D}_{\widetilde{\phi}}$.
Therefore the pullback map $\text{Pic}(\widetilde{B}) \rightarrow
\text{Pic}(\mc{D}_{\widetilde{\phi}})$ is injective.

\medskip\noindent
Finally, suppose that $f-g > g$.  In this case let $(\sigma:C''
\rightarrow \widetilde{B}, \epsilon: \sigma^*(\widetilde{\mc{E}})
\rightarrow \mc{Q}'')$ be the Grassmannian bundle parametrizing rank
$g$ locally free quotients of $\widetilde{\mc{E}}$.  There is an
obvious closed immersion of $C''$ into $\widetilde{B} \times_B C'$.
And the image clearly lies in $\mc{D}_{\widetilde{\phi}}$.  As in the
last paragraph, this implies that the pullback map
$\text{Pic}(\widetilde{B}) \rightarrow
\text{Pic}(\mc{D}_{\widetilde{\phi}})$ is injective.
\end{proof}

\medskip\noindent
As the scheme $\mc{D}_{\widetilde{\phi}} \rightarrow \widetilde{B}$ is
typically not smooth, this is not a log resolution of
$(C',\mc{D}_{\phi})$.  The construction of a log resolution of
$(\widetilde{B} \times_B C', \mc{D}_{\widetilde{\phi}})$ is
essentially equivalent to the construction in Section~\ref{sec-deter}.

\medskip\noindent  
\begin{notat} \label{notat-Co} 
Denote by $C^{(0)}$ the fiber product $C^{(0)} = \widetilde{B}
\times_B C'$ and denote by $M^{(0)} =
M^{(0)}(\widetilde{B},\widetilde{\mc{G}}, \OO_{\widetilde{B}}^{\oplus
g})$ the scheme constructed in Section~\ref{sec-deter}.  Denote by
$p^{(0)}:T^{(0)} \rightarrow C^{(0)}$ the $\text{GL}_g$-torsor
parametrizing sheaf isomorphisms $\text{pr}_2^*\mc{Q}' \rightarrow
\OO_{C^{(0)}}^{\oplus g}$ (with the obvious left $\text{GL}_g$-action)
and denote by $\lambda: (p^{(0)})^* \text{pr}_2^* \mc{Q}' \rightarrow
\OO_{T^{(0)}}^{\oplus g}$ the universal isomorphism.  Denote by
$\epsilon$ the composition of $(p^{(0)})^*\delta:(p^{(0)})^*
\text{pr}_1^* \widetilde{\mc{G}} \rightarrow
(p^{(0)})^*\text{pr}_2^*\mc{Q}'$ with $\lambda$.  Denote by
$q^{(0)}:T^{(0)} \rightarrow M^{(0)}$ the morphism of
$\widetilde{B}$-schemes induced by $\epsilon$.  Observe that this
morphism is equivariant for the obvious $\text{GL}_g$-action on
$M^{(0)}$.  Denote by $g'$ the maximum of $0$ and $2g-f$.
\end{notat}

\begin{lem} \label{lem-qsmooth} 
\begin{enumerate}
\item[(i)]
The image of $q^{(0)}$ equals $M^{(0)} - M^{(0)}_{g'-1}$.
\item[(ii)]
The morphism $q^{(0)}: T^{(0)} \rightarrow \lt( M^{(0)} -
M^{(0)}_{g'-1} \rt)$ factors as an open immersion into a torsor over
$M^{(0)}$ for the vector bundle over $\widetilde{B}$ associated to
$\textit{Hom}_{\OO_{\widetilde{B}}} \lt( \widetilde{\mc{E}},
\OO_{\widetilde{B}}^{\oplus g} \rt)$.  In particular, $q^{(0)}$ is
smooth.
\item[(iii)]
The inverse image scheme $(p^{(0)})^{-1}(\mc{D}_{\widetilde{\phi}})$
equals the inverse image scheme $(q^{(0)})^{-1}(M^{(0)}_{g-1})$.
\end{enumerate}
\end{lem}

\begin{proof}
\textbf{(i)}:
This follows by considering the intersection of the subbundle
$\text{pr}_1^*\widetilde{\mc{G}} \subset \text{pr}_1^* u^*\mc{F}$ with
the kernel of $\text{pr}_1^* u^*\mc{F} \rightarrow
\text{pr}_2^*\mc{Q}'$.  The first subbundle has rank $g$ at every
point, and the second has rank $f-g$.  Hence the maximal possible
intersection is $f-g$ if $f-g \leq g$, and $g$ otherwise. So the
minimal possible rank of $\epsilon$ is $g-(f-g) = 2g-f$ if $2g-f \geq
0$, and $0$ otherwise, i.e. it is $g'$.  On the
other hand, up to composing with an isomorphism $\text{pr}_2^*\mc{Q}'
\rightarrow \OO_{C^{(0)}}^{\oplus g}$, any
morphism $\widetilde{\mc{G}} \rightarrow \OO_{\widetilde{B}}^{\oplus
g}$ can be obtained
as a fiber of $\epsilon$ over a geometric point of $T^{(0)}$.
Therefore $q^{(0)}: T^{(0)} \rightarrow \lt( M^{(0)} - M^{(0)}_{g'-1}
\rt)$ is surjective.

\medskip\noindent
\textbf{(ii)}:
The torsor over $M^{(0)}$ is simply $M^{(0)}(\widetilde{B}, u^*\mc{F},
\OO_{\widetilde{B}}^{\oplus g})$.  The open immersion from $T^{(0)}$
to this scheme is clear.

\medskip\noindent
\textbf{(iii)}
By construction, $\mc{D}_{\widetilde{\phi}}$ is the scheme determined
by the determinant of $\delta$.  And the pullback $(p^{(0)})^*\delta$
equals $(q^{(0)})^* \phi$ by construction.  Therefore the inverse
image of $\mc{D}_{\widetilde{\phi}}$ under $p^{(0)}$ is precisely the
inverse image of $M^{(0)}_{g-1}$ under $q^{(0)}$.
\end{proof}

\medskip\noindent
\begin{notat} \label{notat-Tk} 
Denote by
\begin{equation}
T^{(g)} \rightarrow T^{(g-1)}  \rightarrow \dots \rightarrow T^{(1)}
\rightarrow T^{(0)}
\end{equation}
the sequence of morphisms obtained via base-change by
$q^{(0)}$ from the sequence of morphisms
\begin{equation}
M^{(g)} \rightarrow M^{(g-1)} \rightarrow \dots \rightarrow M^{(1)}
\rightarrow M^{(0)}
\end{equation}
constructed in Lemma~\ref{lem-exist}.
\end{notat}

\medskip\noindent
In particular, each scheme $T^{(k)}$ has a natural
$\text{GL}_g$-action and the each morphism $T^{(k+1)}\rightarrow
T^{(k)}$ is
$\text{GL}_g$-equivariant.  For each $i=1,\dots, g$ the composition
$T^{(i)} \rightarrow T^{(0)}$ is equivalent to the blowing up of an
ideal sheaf $\mc{J}^{(i)}_T$ on $T^{(0)}$.  Moreover, this ideal sheaf
is $\text{GL}_g$-equivariant.  Therefore it is of the form
$(p^{(0)})^{-1} \mc{J}^{(i)}_C$ for some ideal sheaf $\mc{J}{(i)}_C$
on $C^{(0)}$.

\medskip\noindent
\begin{notat} \label{notat-Ck} 
Denote by
\begin{equation}
C^{(g)} \rightarrow C^{(g-1)} \rightarrow \dots \rightarrow C^{(1)}
\rightarrow C^{(0)}
\end{equation}
the sequence of morphisms
obtained by blowing up each of the ideal sheaves $\mc{J}{(i)}_C$.  
\end{notat}

\medskip\noindent
For each $0 \leq r < s \leq g$, there is a Cartesian diagram
\begin{equation}
\begin{CD}
T^{(s)} @>>> T^{(r)} \\
@VVV  @VVV \\
C^{(s)} @>>> C^{(r)}
\end{CD}
\end{equation}
where the vertical arrows are $\text{GL}_g$-torsors, and there is a
Cartesian diagram
\begin{equation}
\begin{CD}
T^{(s)} @>>> T^{(r)} \\
@VVV  @VVV \\
M^{(s)} @>>> M^{(r)}
\end{CD}
\end{equation}
where the vertical arrows are open subsets of torsors for a smooth
group scheme.  By Proposition~\ref{prop-smooth} $M^{(g)} \rightarrow
M^{(0)}$ gives a log resolution of the pair $(M^{(0)},
M^{(0)}_{g-1})$. So $C^{(g)} \rightarrow C^{(0)}$
gives a log resolution of the pair
$(C^{(0)},\mc{D}_{\widetilde{\phi}})$.  Moreover $C^{(0)} \rightarrow
\widetilde{B}$ is smooth.

\medskip\noindent
\begin{notat} \label{notat-Fk} 
Denote by $F_0,\dots, F_{g-1}$ the Cartier divisors on $C^{(g)}$
corresponding to the divisors $E^{(g)}_0, \dots E^{(g)}_{g-1}$ on
$M^{(g)}$.  Of course $F_i = \emptyset$ for $i < g'$.
\end{notat}

\begin{prop}  \label{prop-res2} 
There exists a log resolution $t:C^{(g)} \rightarrow \widetilde{B}
\times_B C'$ of the pair $\lt( \widetilde{B} \times_B
C',\mc{D}_{\widetilde{\phi}} \rt)$ with exceptional locus $F_{g'} \cup
\dots \cup F_{g-2}$ 
satisfying the following properties.
\begin{enumerate}
\item[(i)] 
The morphism $\text{pr}_1 \circ t:C^{(g)} \rightarrow \widetilde{B}$
is smooth, and the intersection of every fiber with $F_{g'} \cup \dots
\cup F_{g-2} \cup F_{g-1}$ is a simple normal crossings divisor.
\item[(ii)] 
The morphism $\text{pr}_2 \circ t:C^{(g)} \rightarrow C'$ is a log
resolution of the pair $(C',\mc{D}_\phi)$ with exceptional locus
$F_{g'} \cup \dots \cup F_{g-2} \cup (\text{pr}_2 \circ t)^{-1} \lt(
E_1 \cup \dots \cup E_k \rt)$ and such that the strict transform of
$\mc{D}_{\phi}$ is the divisor $F_{g-1}$.
\item[(iii)] 
There is an equivalence of $\QQ$-Cartier divisor classes on $C^{(g)}$
\begin{equation}
\begin{array}{l}
K_{C^{(g)}} - (\text{pr}_2 \circ t)^*(K_{C'} + \mc{D}_\phi) = \\
-F_{g-1} + \sum_{j=g'}^{g-2}\lt( (g-1-j)(g-j) -1\rt) F_j + \\
\sum_{i=1}^k
\lt(a(E_i;B,B_{g-1}) - 1 \rt) (\text{pr}_1 \circ t)^* E_i
\end{array}
\end{equation}
\item[(iv)] 
The log discrepancy of $(C',\mc{D}_{\phi})$ equals the minimum of $1$
and the log discrepancy of $(B,B_{g-1})$.  In particular, every
exceptional divisor for $(B,B_{g-1})$ gives rise to an exceptional
divisor for $(C',\mc{D}_{\phi})$.
\item[(v)] 
The pair $(C',\mc{D}_{\phi})$ is log canonical (resp. purely log
terminal, canonical) iff the pair $(B,B_{g-1})$ is log canonical
(resp. purely log terminal, canonical).
\item[(vi)] 
Assume that $\mc{D}_{\phi}$ is normal.  Then the total discrepancy of
$(\mc{D}_{\phi},\emptyset)$ equals the total discrepancy of
$(B,B_{g-1})$.
\item[(vii)] 
Assume that $\mc{D}_{\phi}$ is normal.  Then
$(\mc{D}_{\phi},\emptyset)$ is log canonical (resp. Kawamata log
terminal, canonical) iff the pair $(B,B_{g-1})$ is log canonical
(resp. purely log terminal, canonical).
\item[(viii)] 
Assume that $\mc{D}_{\phi}$ is normal.  If $(B,B_{g-1})$ is terminal,
then $\mc{D}_{\phi}$ is terminal.  If for every exceptional divisor
$E_i$, $a(E_i;B,B_{g-1}) \neq 1$, then the converse also holds.
\end{enumerate}
\end{prop}

\begin{proof}
\textbf{(i)}:
This can be checked after performing the smooth, surjective
base-change by $T^{(g)} \rightarrow C^{(g)}$.  And again by smooth
surjective base-change, the result on $T^{(g)}$ is equivalent to the
statement that $M^{(g)}$ is smooth over $\widetilde{B}$ and every
fiber intersects $E^{(g)}_{g'} \cup \dots \cup E^{(g)}_{g-1}$ in a
simple normal crossings divisor.  This follows from (i) and
(ii) of Proposition~\ref{prop-smooth}.

\medskip\noindent
\textbf{(ii)}:
As mentioned, $t$ is a log resolution of $\mc{D}_{\widetilde{\phi}}$.
Moreover the divisor $F_{g'} \cup \dots F_{g-2}$ is flat over
$\widetilde{B}$ and intersects every fiber in a simple normal
crossings divisor.  Therefore $F_{g'} \cup \dots F_{g-2} \cup
(\text{pr}_2 \circ t)^{-1} \lt( E_1 \cup \dots \cup E_k \rt)$ is a
simple normal crossings divisor.  It follows that $\text{pr}_2\circ t$
is a log resolution of $(C',\mc{D}_{\phi})$. 

\medskip\noindent
\textbf{(iii), (iv) and (v)}:
Item (iii) follows from Corollary~\ref{cor-mults} and
Lemma~\ref{lem-relcan2}.  Items (iv) and (v) follow immediately
from (iii).

\medskip\noindent
\textbf{(vi), (vii) and (viii)}:
Observe that $F_{g-1} \rightarrow
\mc{D}_{\phi}$ is a resolution of singularities.  Also,
\begin{equation}
\begin{array}{l}
K_{F_{g-1}} - (w\circ t)^* K_{\mc{D}_{\phi}} = \\
\sum_{j=g'}^{g-2} \lt( (g-1-j)(g-j) - 1 \rt) F_j|_{F_{g-1}} + \\
\sum_{i=1}^k \lt( a(E_i; B,B_{g-1}) - 1 \rt) (\text{pr}_1 \circ t)^*
E_i.
\end{array}
\end{equation}
By the same sort of argument as in Lemma~\ref{lem-relcan2}, all of the
relevant divisor classes are linearly independent on $F_{g-1}$.  Since
the coefficients $(g-1-j)(g-j)$ are at least $2$ for $j
=g',\dots,g-2$, the total discrepancy of
$\mc{D}_{\phi}$ equals the total discrepancy of $(B,B_{g-1})$.  
This proves (vi).
As
always, it is possible that some exceptional divisors of $(B,B_{g-1})$
do not give rise to exceptional divisors of $\mc{D}_{\phi}$.  
Items (vii) and (viii) follow immediately from (vi).
\end{proof}

\medskip\noindent
\begin{cor} \label{cor-spec} 
Let $S \subset B$ be an irreducible Cartier divisor.  Denote by
$\phi_S$ the restriction of $\phi$ to $S$.  Assume that
$\mc{D}_{\phi_S}$ is irreducible.
\begin{enumerate}
\item[(i)]
Suppose that $S$ is Kawamata log terminal.  Then $(S,S_{g-1})$ is log
canonical iff there exists an open subscheme $U\subset B$ containing
$S$ such that $(U,S+U_{g-1})$ is log canonical.  In particular, if
$(S,S_{g-1})$ is log canonical, then $(U,U_{g-1})$ is log canonical.
\item[(ii)] 
Suppose that $\mc{D}_{\phi_S}$ is irreducible and normal. Then
$(S,S_{g-1})$ is Kawamata log terminal iff there exists an open
subscheme $U\subset B$ containing $S$ such that $(U,S+U_{g-1})$ is
purely log terminal.  In particular, if $(S,S_{g-1})$ is Kawamata log
terminal, then $(U,U_{g-1})$ is Kawamata log terminal.
\item[(iii)] 
Suppose that $B$ is Gorenstein and that $\mc{D}_{\phi_S}$ is
irreducible and normal.  Then $(S,S_{g-1})$ is canonical iff there
exists an open subscheme $U\subset B$ containing $S$ such that
$(U,U_{g-1})$ is canonical.  In particular, if $(S,S_{g-1})$ is
canonical then $(U,U_{g-1})$ is canonical.
\end{enumerate}
\end{cor}

\medskip\noindent
\begin{proof}
\textbf{(i)}:
First observe that $\rho^{-1}(S) \subset C'$ is also
Kawamata log terminal since $\rho$ is smooth.  By
Proposition~\ref{prop-res2}, $(S,S_{g-1})$ is log canonical iff
$(\rho^{-1}(S), \mc{D}_{\phi_S})$ is log canonical.  By a similar
argument, $(B,S+B_{g-1})$ is log canonical iff
$(C',\rho^{-1}(S)+\mc{D}_{\phi})$ is log canonical.  By
\cite[Thm. 7.5.2]{KollarPairs}, $(\rho^{-1}(S),\mc{D}_{\phi_S})$ is
log canonical iff $(C',\rho^{-1}(S) + \mc{D}_{\phi})$ is log canonical
near $\rho^{-1}(S)$.  Therefore $(S,S_{g-1})$ is log canonical iff
$(B,S+B_{g-1})$ is log canonical near $S$. 

\medskip\noindent
\textbf{(ii) and (iii)}:
For (ii), combine (vii) of
Proposition~\ref{prop-res2} with ~\cite[Thm. 7.5.1]{KollarPairs}.  For
(iii), combine (vii) of Proposition~\ref{prop-res2}
with ~\cite{Stevens88} (see also ~\cite[Thm. 7.9]{KollarPairs}).
\end{proof}

\begin{rmk} \label{rmk-spec} 
If one further assumes that $B$ is a local complete intersection
scheme, then one can also use the results of Ein and Musta\c{t}\v{a}
~\cite{EM} to prove Corollary~\ref{cor-spec} and to relate the minimal
log discrepancy of $(S,S_{g-1})$ to the minimal log discrepancy of
$(B,B_{g-1})$.
\end{rmk}

\section{Deformation to the normal cone} \label{sec-dnc}
\medskip\noindent
Corollary~\ref{cor-spec} concludes results about
$(B,B_{g-1})$ by analyzing the restriction of $\phi$ to an irreducible
Cartier divisor $S$.  In applications it is also natural to restrict
$\phi$ to an irreducible closed subvariety $Y\subset B$ that is not
necessarily a Cartier divisor; in particular, if $r$ is the smallest
integer such that $B_r \neq \emptyset$, then one natural choice is to
take $Y$ to be an irreducible component of $B_r$.  In this section,
the general case is reduced to the case of a Cartier divisor
by using deformation to the normal cone.  We briefly review the
discussion of deformation to the normal cone from ~\cite[Chapter
5]{F}.  All of the unproved assertions regarding deformation to the
normal cone used here are proved there.

\medskip\noindent
The following setup is a little more general than needed. 
Let
$\phi:\mc{G} \rightarrow \mc{F}$ be a morphism of locally free
$\OO_X$-modules of ranks $g$ and $f$
(but do not assume that $f > g$).  Denote by $\mc{E}$ the cokernel of
$\phi$.
Let $Y \subset B$ be a closed subscheme with ideal sheaf
$\mc{J}$.  Denote by $\mc{K}_Y$ and $\mc{E}_Y$ the kernel and cokernel
respectively of the map of $\OO_Y$-modules
\begin{equation}
\phi\otimes \text{Id}: \mc{G} \otimes_{\OO_B} \OO_Y \rightarrow \mc{F}
\otimes_{\OO_B} \OO_Y.
\end{equation}
In particular $\mc{E}_Y$ is simply $\mc{E} \otimes_{\OO_B} \OO_Y$.
Consider the following commutative diagram with exact rows
\begin{equation} \label{eqn-snake1} 
\begin{CD}
0 @>>> \mc{G}\otimes_{\OO_B} \mc{J}/\mc{J}^2 @>>> \mc{G} \otimes_{\OO_B}
 \OO_B/\mc{J}^2 @>>> \mc{G} \otimes_{\OO_B} \OO_Y @>>> 0 \\
& & @V \phi_1 VV @VV\phi_2 V @VV\phi_3 V \\
0 @>>> \mc{F}\otimes_{\OO_B} \mc{J}/\mc{J}^2 @>>> \mc{F} \otimes_{\OO_B}
 \OO_B/\mc{J}^2 @>>> \mc{F} \otimes_{\OO_B} \OO_Y @>>> 0 
\end{CD}
\end{equation}
Each map $\phi_i$ is just $\phi \otimes \text{Id}$.  Since tensor
product is right exact, the cokernel of $\phi_1$ is just $\mc{E}_Y
\otimes_{\OO_Y} \mc{J}/\mc{J}^2$.  And by definition the kernel of $\phi_3$ is
$\mc{K}_Y$.  By the Snake Lemma, there is an induced connecting map
from $\text{Ker}(\phi_3)$ to $\text{Coker}(\phi_1)$.

\medskip\noindent
\begin{defn} \label{defn-conn} 
The \emph{connecting map}, denoted $\theta = \theta_{\phi,Y}: \mc{K}_Y
\rightarrow \mc{E}_Y \otimes_{\OO_Y} \mc{J}/\mc{J}^2$, is the map of
$\OO_Y$-modules induced by the Snake Lemma as above.  The
\emph{induced map} of the connecting map is the map $\theta'_{\phi,Y}:
\textit{Hom}_{\OO_Y}(\mc{E}_Y,\mc{K}_Y) \rightarrow \mc{J}/\mc{J}^2$
induced from $\theta_{\phi,Y}$ by adjointness of Hom and tensor
product.
\end{defn}

\medskip\noindent
\begin{hyp} \label{hyp-conn} 
From now on, $\mc{E}_Y$ is assumed to be a locally free
$\OO_Y$-module.  This 
implies that also $\mc{K}_Y$ is a locally free $\OO_Y$-module. 
In the
later sections, $\mc{J}/\mc{J}^2$ will also be a
locally free $\OO_Y$-module
(but this is \emph{not} a hypothesis in this section).
\end{hyp}

\begin{lem} \label{lem-conn} 
\begin{enumerate}
\item[(i)] 
For the transpose $\phi^\dagger$, the kernel of $\phi^\dagger|_Y$ is
$\mc{E}_Y^\vee$, the cokernel of $\phi^\dagger|_Y$ is $\mc{K}_Y^\vee$,
and the connecting map $\theta'_{\phi^\dagger,Y}$ is identified with
$\theta'_{\phi,Y}$ under the canonical isomorphism
$$ \textit{Hom}_{\OO_Y}(\mc{K}^\vee_Y,\mc{E}^\vee_Y) \cong
  \textit{Hom}_{\OO_Y}(\mc{E}_Y, \mc{K}_Y).
$$
\item[(ii)] 
Let $\phi':\mc{G}' \rightarrow \mc{F}'$ be a second morphism of
locally free sheaves on $B$ such that the kernel and cokernel of
$\phi'|_Y$ are locally free $\OO_Y$-modules, 
$\mc{K}'_Y$ and $\mc{E}'_Y$.  Consider
$\phi\oplus \phi': \mc{G} \oplus \mc{G}' \rightarrow \mc{F} \oplus
\mc{F}'$.  The kernel of $(\phi \oplus \phi')|_Y$ is $\mc{K}_Y \oplus
\mc{K}'_Y$, the cokernel of $(\phi \oplus \phi')|_Y$ is $\mc{E}_Y
\oplus \mc{E}'_Y$, and $\theta_{\phi\oplus \phi',Y}$ equals
$\theta_{\phi,Y} \oplus \theta_{\phi',Y}$ via the canonical
isomorphisms.
\item[(iii)] 
Consider $\phi\otimes \phi':\mc{G}
\otimes_{\OO_B} \mc{G}' \rightarrow \mc{F} \otimes_{\OO_B} \mc{F}'$.
The kernel of $\phi\otimes \phi'$ is the surjective image of $\lt(
\mc{K}_Y \otimes_{\OO_Y} \mc{G}'|_Y \rt) \oplus \lt( \mc{G}|_Y
\otimes_{\OO_Y} \mc{K}'_Y \rt)$, the cokernel of $\phi \otimes \phi'$
is a subsheaf of $\lt( \mc{E}_Y \otimes_{\OO_Y} \mc{F}'|_Y \rt) \oplus
\lt( \mc{F}|_Y \otimes_{\OO_Y} \mc{E}'_Y \rt)$, and
$\theta_{\phi\otimes \phi',Y}$ is the unique morphism compatible with
$( \theta_{\phi,Y} \otimes \phi' ) \oplus (\phi \otimes
\theta_{\phi',Y})$.
\item[(iv)] 
Let $\psi_{\mc{G}}:\mc{G}
\rightarrow \mc{G}'$ and $\psi_{\mc{F}}:\mc{F} \rightarrow \mc{F}'$ be
morphisms of $\OO_B$-modules such that there is a commutative diagram:
\begin{equation}
\begin{CD}
\mc{G} @> \phi >> \mc{F} \\
@V \psi_{\mc{G}} VV @VV \psi_{\mc{F}} V \\
\mc{G}' @> \phi'>> \mc{F}' 
\end{CD}
\end{equation}
There are unique morphisms $\psi_{K}:\mc{K}_Y \rightarrow \mc{K}'_Y$
and $\psi_{E}:\mc{E}_Y \rightarrow \mc{E}'_Y$ such that the following
diagram commutes:
\begin{equation}
\begin{CD}
0 @>>> \mc{K}_Y @>>> \mc{G}|_Y @> \phi|_Y >> \mc{F}|_Y @>>> \mc{E}_Y
@>>> 0 \\
& & @V \psi_{\mc{K}} VV @V \psi_{\mc{G}} VV @VV \psi_{\mc{F}} V @VV
\psi_{\mc{E}} V \\
0 @>>> \mc{K}'_Y @>>> \mc{G}'|_Y @> \phi'|_Y >> \mc{F}'|_Y @>>> \mc{E}'_Y
@>>> 0 \\
\end{CD}
\end{equation}
Moreover, the diagram of connecting maps commutes:
\begin{equation}
\begin{CD}
\mc{K}_Y @> \theta_{\phi,Y} >> \mc{E}_Y \otimes_{\OO_Y}
\mc{J}/\mc{J}^2 \\
@V \psi_{\mc{K}} VV @VV \psi_{\mc{E}}\otimes \text{Id} V \\
\mc{K}'_Y @> \theta_{\phi',Y} >> \mc{E}'_Y \otimes_{\OO_Y}
\mc{J}/\mc{J}^2. 
\end{CD}
\end{equation}
\end{enumerate}
\end{lem}

\begin{proof}
Each of these follows by simple diagram-chasing.  The details are
left to the reader.
\end{proof}

\medskip\noindent
Deformation to the normal cone,
which is described in detail in ~\cite[Chapter 5]{F}, is as follows.  
Form the product $B \times
\PP^1$ and consider the closed subscheme $Y \times \{\infty \} \subset
B \times \PP^1$.  The ideal sheaf of this subscheme is
\begin{equation}
\mc{J}' = 
\text{pr}_1^{-1}(\mc{J}) +
\text{pr}_2^{-1}(\OO_{\PP^1}(-\underline{\infty})).
\end{equation}
This decomposition of the ideal sheaf yields a decomposition of the
Rees algebra
\begin{equation}
\begin{array}{l}
\oplus_{n=0}^\infty (\mc{J}')^n/(\mc{J}')^{n+1} \cong   \\
\text{pr}_1^*\lt( \oplus_{n=0}^\infty (\mc{J})^n/ \mc{J}^{n+1} \rt)
\otimes \text{pr}_2^* \lt( \oplus_{n=0}^\infty
\OO_{\PP^1}(-n\underline{\infty})/
\OO_{\PP^1}(-(n+1)\underline{\infty}) \rt).
\end{array}
\end{equation}
The relative Spec of the Rees algebra is the \emph{normal cone}.  
If the
normal cone is the symmetric algebra of a locally free sheaf, the
normal cone is called the \emph{normal bundle}.  The
decomposition above gives an isomorphism of the normal cone
$C_{Y\times\{ \infty \} }(B\times \PP^1)$ with the fiber product
$\text{pr}_1^* C_{Y} B \times_{Y \times \{ \infty \} } \text{pr}_2^*
N_{\{\infty\} } \PP^1$.

\begin{notat} \label{notat-cones}
Denote $C= \text{pr}_1^* C_Y B$ and $C'= C_{Y
\times \{\infty \} } (B \times \PP^1)$.  The normal bundle
$N_{\{ \infty \} } \PP^1$ is just the trivial rank $1$ vector bundle,
denoted by $\mathbf{1}$ in ~\cite{F}.  
\end{notat}

\medskip\noindent
The isomorphism of algebras
respects the $\mathbb{G}_m$-actions induced by the grading of the
algebras.  Therefore, in the notation of ~\cite{F}, there is an
equivalence of cones $C' \cong C \oplus \mathbf{1}$.

\medskip\noindent
Let $u:M \rightarrow B \times \PP^1$ be the blowing up of $B
\times \PP^1$ along $Y \times \{ \infty \}$.  Denote by $\varrho: M
\rightarrow \PP^1$ the composition $\text{pr}_2 \circ u$.  This is a
flat morphism.  The preimage of $\AAA^1 = \PP^1 - \{ \infty \}$ is
isomorphic to $B \times \AAA^1$ (compatible with the projections to $B$ and
to $\AAA^1$).  And the Cartier divisor $M_{\infty} =
\varrho^{-1}(\infty)$ is the sum of two effective divisors $B_Y$ and
$\PP(C') = \PP(C \oplus \mathbf{1})$.  Here $B_Y$ is the blowing up of
$B$ along $Y$.  And, as usual, for a cone $K$, the symbol $\PP(K)$
means the relative Proj of the graded algebra associated to $K$.
Denote by $\pi: \PP(C \oplus \mathbf{1}) \rightarrow Y$ the obvious
projection morphism.

\medskip\noindent
The intersection of $B_Y$ and $\PP(C \oplus \mathbf{1})$ is the
exceptional divisor on $B_Y$ and is the ``hyperplane section at
infinity'' $\PP(C)$ in $\PP(C \oplus \mathbf{1})$.  The complement of
the hyperplane section at infinity is identified with the cone $C$
over $Y$.  Finally, there is a closed immersion $\iota: Y \times \PP^1
\rightarrow M$ such that $u\circ \iota: Y \times \PP^1 \rightarrow B
\times \PP^1$ is the obvious closed immersion.  The fiber of $\iota(Y
\times \PP^1)$ over $\infty$ is identifed with the zero section of $C
\subset \PP(C \oplus \mathbf{1})$.

\begin{defn} \label{defn-dnc} 
For a closed subscheme $Y \subset B$, the \emph{deformation to the
normal cone} is the datum $(\varrho: M \rightarrow \PP^1, \iota: Y
\times \PP^1 \hookrightarrow M, B_Y \hookrightarrow M, \PP(C\oplus
\mathbf{1}) \hookrightarrow M)$.  Denote by $\phi_M: \mc{G}_M
\rightarrow \mc{F}_M$ the morphism of locally free sheaves
$u^*\text{pr}_1^* \phi$.
\end{defn}

\medskip\noindent
On $\PP(C\oplus \mathbf{1})$ there is a rank $1$ locally free quotient
$\beta: \pi^*\lt( \text{pr}_1^* \mc{J}/\mc{J}^2 \oplus \OO_Y \rt)
\rightarrow \OO_{\PP(C \oplus \mathbf{1})}(1)$ (satisfying a well-known
universal property).  Denote by $\beta_1:
\pi^* \lt( \text{pr}_1^* \mc{J}/\mc{J}^2 \rt) \rightarrow \OO_{\PP(C
\oplus \mathbf{1})}(1)$ and $\beta_2: \OO_{\PP(C\oplus \mathbf{1})}
\rightarrow \OO_{\PP(C \oplus \mathbf{1})}(1)$ the two components of
$\beta$.  Of course, the zero scheme of the section $\beta_2$ is
precisely the hyperplane section at infinity $\PP(C) \subset \PP(C
\oplus \mathbf{1})$.  The invertible sheaf $\OO_M(-\PP(C\oplus
\mathbf{1}))|_{\PP(C\oplus \mathbf{1})}$ is canonically isomorphic to
$\OO_{\PP(C \oplus \mathbf{1})}(1)$; the isomorphism is induced by the
isomorphism of ideal sheaves $\text{u}^{-1}\mc{J}' \cong \OO_M(-\PP(C
\oplus \mathbf{1}))$.

\medskip\noindent 
The pullback $\phi_M$ factors through an
\emph{elementary-transform-up} 
of $\mc{G}_M$.  To describe this elementary-transform-up, first 
dualize all sheaves and form the adjoints of all sheaf morphisms.  
Consider the adjoint morphism $\phi_M^\dagger:
\mc{F}_M^\vee \rightarrow \mc{G}_M^\vee$.  The restriction of $\phi_M$
to $\PP(C\oplus \mathbf{1})$ is just $\pi^*(\phi|_Y)$.  In particular,
the image of $\lt( \mc{F}_M^\vee \rt)|_{\PP(C \oplus \mathbf{1})}$ is
contained in the kernel of $\pi^*\lt( \mc{G} \otimes_{\OO_B} \OO_Y
\rt)^\vee \rightarrow \pi^*\lt( \mc{K}_Y \rt)^\vee$.  Define the
subsheaf $(\widetilde{\mc{G}} )^\vee \subset \lt( \mc{G}_M \rt)^\vee$
to be the kernel of the surjection
\begin{equation}
\lt( \mc{G}_M \rt)^\vee \rightarrow \lt( \mc{G}_M
\rt)^\vee|_{\PP(C\oplus \mathbf{1})} \cong \pi^*\lt( \mc{G}
\otimes_{\OO_B} \OO_Y \rt)^\vee \rightarrow \pi^* \lt( \mc{K}_Y
\rt)^\vee.
\end{equation}
Then $\phi_M^\dagger$ factors through the subsheaf
$(\widetilde{\mc{G}})^\vee$.  Define $( \widetilde{\phi} )^\dagger:
\mc{F}_M^\vee \rightarrow ( \widetilde{\mc{G}} )^\vee$ to be the
induced map.

\medskip\noindent
\begin{lem} \label{lem-elemtrans} 
Denote by $\widetilde{\mc{G}}$ the dual of
$(\widetilde{\mc{G}})^\vee$, and denote by $\widetilde{\phi}$ the
adjoint of $(\widetilde{\phi})^\dagger$.
\begin{enumerate}
\item[(i)]
The sheaf $(\widetilde{\mc{G}} )^\vee$ is locally free of rank $g$.
Therefore also $\widetilde{\mc{G}}$ is locally free of rank $g$.
\item[(ii)]
The cokernel of the 
sheaf map $\mc{G}_M
\rightarrow \widetilde{\mc{G}}$ is canonically isomorphic to the
push-forward from $\PP(C\oplus \mathbf{1})$ of the locally free sheaf
$\pi^* \mc{K}_Y \otimes \OO_{\PP(C \oplus \mathbf{1})}(-1)$.  
\item[(iii)]
The restriction of $\mc{G}_M \rightarrow \widetilde{\mc{G}}$ to $\PP(C
\oplus \mathbf{1})$ fits into an exact sequence
\begin{equation}
\begin{array}{lllllll}
0 &  \longrightarrow & \pi^* \mc{K}_Y & \longrightarrow & \pi^*(\mc{G}
 \otimes_{\OO_B} \OO_Y ) & \longrightarrow \\
 & &
\widetilde{\mc{G}}|_{\PP(C \oplus \mathbf{1})} & \longrightarrow &
 \pi^* \mc{K}_Y 
\otimes \OO_{\PP(C \oplus \mathbf{1})}(-1)
& \longrightarrow & 0
\end{array}
\end{equation}
\end{enumerate}
\end{lem}

\begin{proof}
Item (i) is very easy and is just the fact that an 
\emph{elementary-transform-down} along a Cartier divisor gives rise to
a locally free 
sheaf.  For (ii) and (iii), observe that the restriction
to $\PP(C \oplus \mathbf{1})$ of $(\widetilde{\mc{G}})^\vee
\rightarrow (\mc{G}_M)^\vee$ fits into an exact sequence,
\begin{equation}
\begin{array}{lllllll}
0 & \longrightarrow & \textit{Tor}_1^{\OO_M}(\OO_{\PP(C \oplus
  \mathbf{1})}, \pi^*\mc{K}^\vee_Y) & \longrightarrow &
  (\widetilde{\mc{G}})^\vee|_{\PP(C \oplus \mathbf{1})} \\
 & \longrightarrow 
  &  \pi^*( \mc{G} \otimes_{\OO_B} \OO_Y )^\vee & \longrightarrow &
  \pi^* \mc{K}^\vee_Y & \longrightarrow & 0.
\end{array}
\end{equation}
Of course $\textit{Tor}_1^{\OO_M}(\OO_{\PP(C \oplus \mathbf{1})},
\OO_{\PP(C \oplus \mathbf{1})} )$ is just $\OO_M(-\PP(C \oplus
\mathbf{1}))|_{ \PP(C \oplus \mathbf{1}) }$, i.e. $\OO_{\PP(C \oplus
\mathbf{1})}(1)$.  Therefore the left-most term in the exact sequence
is just $\mc{K}^\vee_Y \otimes \OO_{ \PP(C \oplus \mathbf{1})
}(1)$.  Dualizing this sequence gives (ii) and (iii).
\end{proof}

\medskip\noindent 
\begin{notat} \label{notat-gamma} 
The restriction of $\widetilde{\phi}$ to $\PP(C \oplus \mathbf{1})$
induces a morphism of locally free sheaves
\begin{equation}
\widetilde{\mc{G}}|_{\PP(C\oplus
  \mathbf{1})}/\pi^*(\mc{G}\otimes_{\OO_B} \OO_Y) \rightarrow
  \pi^*(\mc{F} \otimes_{\OO_B} \OO_Y)/\pi^*(\mc{G} \otimes_{\OO_B}
  \OO_Y).
\end{equation}
Up to canonical isomorphism, this is the same as a morphism
\begin{equation}
\gamma: \pi^*\mc{K}_Y \otimes \OO_{\PP(C \oplus \mathbf{1})}(-1)
\rightarrow \pi^* \mc{E}_Y.
\end{equation}
And the cokernel of $\widetilde{\phi}$ on $\PP(C \oplus \mathbf{1} )$
equals the cokernel of $\gamma$.
\end{notat}

\medskip\noindent
The next goal is to show that the map $\gamma$ essentially is the same
as the map $\theta_{\phi,Y}$.

\medskip\noindent
The canonical inclusion $\widetilde{\mc{G}}\otimes \OO_M(-\PP(C \oplus
\mathbf{1}) ) \rightarrow \widetilde{\mc{G}}$ factors through the
kernel of $\widetilde{\mc{G}} \rightarrow \pi^*\mc{K}_Y \otimes
\OO_{\PP(C \oplus \mathbf{1})}(-1)$.  So there is an induced inclusion
$\widetilde{\mc{G}} \otimes \OO_M(-\PP(C \oplus \mathbf{1}))
\rightarrow \mc{G}_M$ whose cokernel is $\pi^*\lt( \mc{G}
\otimes_{\OO_B} \OO_Y/ \mc{K}_Y \rt)$.  In particular, there is a
commutative diagram:
\begin{equation} \label{eqn-cd1}
\begin{CD}
\widetilde{\mc{G}} \otimes \OO_M(-\PP(C \oplus \mathbf{1}) ) @>>>
\mc{G}_M \\
@V \widetilde{\phi} \otimes \text{Id} VV @VV \phi_M V \\
\mc{F}_M \otimes \OO_M(-\PP(C \oplus \mathbf{1}) ) @>>> \mc{F}_M
\end{CD}
\end{equation}

\medskip\noindent
\begin{lem} \label{lem-thetagamma} 
There is a commutative diagram of coherent sheaves:
\begin{equation} 
\begin{CD}
\pi^*\mc{K}_Y @> \pi^*\theta_{\phi,Y} >> \pi^*\mc{E}_Y \otimes
\pi^*(\mc{J}/\mc{J}^2) \\
@V \text{Id} VV @VV \text{Id} \otimes \beta_1 V \\
\pi^*\mc{K}_Y @> \gamma \otimes \text{Id} >> \pi^* \mc{E}_Y \otimes
\OO_{\PP(C \oplus \mathbf{1})}(1)
\end{CD}
\end{equation}
\end{lem}

\medskip\noindent
\begin{proof}
To ease notation in this proof, denote $\PP = \PP(C \oplus
\mathbf{1})$.  Consider the commutative diagram with exact rows
analogous to Equation(~\ref{eqn-snake1}) whose rows are
\begin{equation} \label{eqn-snake2a}
\begin{array}{ccccc}
0 & \longrightarrow & \mc{G}_M \otimes_{\OO_M} \OO_M(-\PP)/ 
\OO_M(-2\PP) & \longrightarrow & \mc{G}_M \otimes_{\OO_M}
\OO_M/ \OO_M(-2\PP) \\
 & \longrightarrow & \mc{G}_M
\otimes_{\OO_M} \OO_{\PP} & \longrightarrow & 0 
\end{array}
\end{equation} 

\begin{equation} \label{eqn-snake2b}
\begin{array}{ccccc}
0 & \longrightarrow & \mc{F}_M \otimes_{\OO_M} \OO_M(-\PP)/ 
\OO_M(-2\PP) & \longrightarrow & \mc{F}_M \otimes_{\OO_M}
\OO_M/ \OO_M(-2\PP) \\
 & \longrightarrow & \mc{F}_M
\otimes_{\OO_M} \OO_{\PP} & \longrightarrow & 0 
\end{array}
\end{equation} 
Associated to this commutative diagram, the snake lemma produces a
connecting map 
$\theta_{\phi_M,\PP}:\pi^* \mc{K}_Y \rightarrow \pi^*\mc{E}_Y \otimes
\OO_{\PP}(1)$.  

\medskip\noindent
Observe that the ideal sheaf $u^{-1}\text{pr}_1^{-1}(\mc{J})$ is
contained in the ideal sheaf $\OO_M(-\PP)$ (moreover, after dividing by
the defining equation of $\PP$, the residual ideal sheaf is the ideal
sheaf of the closed immersion $\iota:Y \times \PP^1 \rightarrow M$).
Therefore there is a map from the pullback by $\text{pr}_1 \circ u$ of
the commutative diagram in Equation(~\ref{eqn-snake1}) to the
commutative diagram above.  In particular there is a commutative
diagram of connecting maps
\begin{equation}
\begin{CD}
\pi^* \mc{K}_Y @> \pi^* \theta_{\phi, Y} >> \pi^* \mc{E}_Y \otimes
\pi^*\lt( \mc{J}/\mc{J}^2 \rt) \\
@V \text{Id} VV @VV \text{Id} \otimes \beta V \\
\pi^* \mc{K}_Y @> \theta_{\phi_M, \PP} >> \pi^* \mc{E}_Y \otimes
\OO_{\PP}(1)
\end{CD}
\end{equation}
Now consider the map $\widetilde{\mc{G}}(-\PP) \rightarrow
\mc{G}_M \otimes_{\OO_M} \OO_M/\OO_M(-2\PP)$ constructed above.  The
image of $\widetilde{\mc{G}}(-\PP)$ in the quotient $\mc{G}_M
\otimes_{\OO_M} \OO_{\PP}$ is precisely $\pi^* \mc{K}_Y$.  Moreover
there is the map $\widetilde{\phi} \otimes \text{Id}:
\widetilde{\mc{G}}(-\PP) \rightarrow \mc{F}_M(-\PP)$ and the diagram
in Equation (~\ref{eqn-cd1}) commutes.  Therefore 
$\widetilde{\mc{G}}(-\PP)$ can be used
to compute the connecting map
$\theta_{\phi_M,\PP}$.  But the construction of $\gamma$ was by
precisely the same construction, i.e.  the connecting map
$\theta_{\phi_M,\PP}$ equals $\gamma\otimes \text{Id}$.  
\end{proof}

\section{The stack of multiple covers of lines} \label{sec-covers}
\medskip\noindent

\begin{hyp} \label{hyp-char0}
From this point on, the base field $K$ is algebraically
closed of characteristic $0$.  
\end{hyp}

Let $V$ denote a $K$-vector space of
dimension $n+1$ so that the projective space $\PP(V)$ is isomorphic to
$\PP^n$.  In this and the next sections, results from the previous
sections are applied to
$\Kbm{0,0}{\PP(V),e}$, the Kontsevich moduli stack parametrizing
stable maps from unmarked, genus $0$ curves to $\PP(V)$ of degree $e$.
For more details about this stack, see ~\cite{FP}.  The goal is to
prove that for a positive integer $d$ with $d+e \leq n$, for a general
hypersurface $X\subset \PP(V)$ of degree $d$, the closed substack
$\Kbm{0,0}{X,e} \subset \Kbm{0,0}{\PP(V),e}$ has at worst canonical
singularities.

\begin{rmk} \label{rmk-DMsings}
What does it mean to say that a pair of Deligne-Mumford stacks is
canonical (resp. log canonical, etc.)?  For a pair $(B,Y)$, one can
compute the log discrepancy $a(E;B,Y)$ \'etale locally on $B$, i.e. if
$(f_i:B_i \rightarrow B)$ is an \'etale cover, then $a(E;B,Y) =
\text{min}(a(f_i^* E; B_i, f_i^{-1}Y) | \text{center}(f_i^* E) \neq
\emptyset)$.  There is a standard way of extending any \'etale local
notion for schemes to Deligne-Mumford stacks: the Deligne-Mumford
stack has an \'etale local cover by schemes and the log discrepancies
are defined using this cover by the formula above.
\end{rmk}

\medskip\noindent
Consider the Kontsevich moduli stack $\Kbm{0,r}{\PP(V),e}$.  Let
$p:\mc{C} \rightarrow \Kbm{0,r}{\PP(V),e}$ denote the universal curve,
and let $f:\mc{C} \rightarrow \PP(V)$ denote the universal map.  For
each integer $d> 0$, on $\Kbm{0,r}{\PP(V),e}$ there is a locally free
sheaf $\mc{P}_d$ of rank $ed+1$ defined by $\mc{P}_d = p_* f^*
\OO_{\PP(V)}(d)$.  
Standard facts about stable maps together with cohomology and base
change imply 
that the higher direct images of $f^* \OO_{\PP(V)}(d)$ vanish and
that $\mc{P}_d$ is locally free of rank $ed+1$.
The case of interest is $r=0$, but
in fact this holds for arbitrary $r$.  There is a canonical
\emph{evaluation morphism} of locally free sheaves on
$\Kbm{0,0}{\PP(V),e}$, $\phi_d^\dagger: H^0(\PP(V),\OO_{\PP(V)}(d))
\otimes_\kappa \OO \rightarrow \mc{P}_d$.

\medskip\noindent
\begin{defn} \label{defn-GtoF} 
For each $e \geq 1$ and $d\geq 0$, define $\mc{G}_d$ to be the dual
of $\mc{P}_d$, define $\mc{F}_d$ to be the trivial locally free
sheaf $H^0(\PP(V),\OO_{\PP(V)}(d))^\vee \otimes_\kappa \OO$, and 
define the \emph{co-evaluation morphism for degree $d$} to be the morphism,
\begin{equation}
\phi_d: \mc{G} \rightarrow \mc{F},
\end{equation}
adjoint to the evaluation morphism.  Define
$\mc{E}_d$ to be the cokernel of $\phi_d$.
Define $(\pi_d:C_d \rightarrow \Kbm{0,0}{\PP(V),e}, \alpha_d: \pi^*
\mc{E}_d \rightarrow \mc{Q}_d)$ to be the projective Abelian cone
parametrizing rank $1$ locally free quotients of $\mc{E}_d$.
When there is no risk of confusion, the subscripts will be dropped.  
\end{defn}

\medskip\noindent
The goal is to analyze the singularities of $C$.  
Denote by $(\rho:C' \rightarrow \Kbm{0,0}{\PP(V),e},\beta:
H^0(\PP(V),\OO_{\PP(V)}(d))^\vee \otimes_\kappa \OO \rightarrow
\mc{Q}')$ the projective bundle parametrizing rank $1$ locally free
quotients of the trivial locally free sheaf
$H^0(\PP(V),\OO_{\PP(V)}(d)) \otimes_\kappa \OO$.  Of course $C'$ is 
the same as the product $\PP H^0(\PP(V),\OO_{\PP(V)}(d)) \times
\Kbm{0,0}{\PP(V),e}$.  As in Section~\ref{sec-discrep}, define $h:C
\rightarrow C'$ to be the tautological closed immersion.  Our interest
in $C$ comes from the following easy result.  
The proof is left to the reader
(but also see ~\cite[Lemma 4.5]{HRS2}).

\medskip\noindent
\begin{lem} \label{lem-PE} 
The Deligne-Mumford stack $C$ parametrizes pairs $([X],[f:C\rightarrow
X])$ consisting of 
$[X]\in \PP H^0(\PP(V),\OO_{\PP(V)}(d))$, a hypersurface
of degree $d$ in $\PP(V)$, and $f:C\rightarrow X$, a
Kontsevich stable map of genus $0$ and degree $e$ to $X$.  In
particular, for each $[X] \in \PP H^0(\PP(V),\OO_{\PP(V)}(d))$, the
fiber of $C$ over $[X]$ is canonically identified with the Kontsevich
moduli stack $\Kbm{0,0}{X,e}$.
\end{lem}

\medskip\noindent
\begin{notat} \label{notat-Grass} 
Denote by $\mathbb{G} = \mathbb{G}(2,V)$ the Grassmannian variety over
$\kappa$ parametrizing rank $2$ locally free quotients of $V^\vee$,
i.e. parametrizing $2$-dimensional linear subspaces of $V$.  Let
$V^\vee \otimes_\kappa \OO_{\mathbb{G}} \rightarrow S^\vee$ denote the
universal quotient, so that the adjoint $S \hookrightarrow V
\otimes_\kappa \OO_{\mathbb{G}}$ is the universal rank $2$ subbundle.
Denote the quotient of the universal subbundle by $V \otimes_\kappa
\OO_{\mathbb{G}} \rightarrow T$.  Observe that $T^\vee \hookrightarrow
V^\vee \otimes_\kappa \OO_{\mathbb{G}}$ is simply the annihilator of
$S$.  For each $d \geq 0$, there is an induced filtration on
$\text{Sym}^d(V^\vee) \otimes_\kappa \OO_{\mathbb{G}}$
\begin{equation}
\begin{array}{l}
\text{Sym}^d(V^\vee) \otimes_\kappa \OO_{\mathbb{G}} = F^0 \supset F^1
\supset \dots \supset F^d \\
F^i = \text{Sym}^i(T^\vee) \cdot \text{Sym}^{d-i}(V^\vee), \ \ \
F^i/F^{i+1} \cong \text{Sym}^i(T^\vee) \otimes_{\OO_{\mathbb{G}}}
\text{Sym}^{d-i}(S^\vee)
\end{array}
\end{equation}
This filtration is the same as the filtration by order of vanishing
along $S$.
\end{notat}


\medskip\noindent
\begin{defn} \label{defn-covers} 
There is an induced morphism $\PP S \rightarrow \PP(V)$ identifying
$\mathbb{G}$ with the Hilbert scheme of lines in $\PP(V)$.  The
\emph{stack of multiple covers of lines} $Y$ is defined to be the
closed substack of $\mathbb{G} \times \Kbm{0,0}{\PP(V),e}$
parametrizing pairs $([L],[f:C \rightarrow L])$ where $[L]\in
\mathbb{G}$ is a line in $\PP(V)$ and $f:C\rightarrow L$ is a
Kontsevich stable map of genus $0$ and degree $e$.
\end{defn}

\medskip\noindent 
There are several equivalent definitions.  Of course the projection
$\text{pr}_{\mathbb{G}}: Y \rightarrow \mathbb{G}$ is Zariski locally
isomorphic to the product $\mathbb{G} \times \Kbm{0,0}{\PP^1,e}$.  An
easy observation is that the projection $Y \rightarrow
\Kbm{0,0}{\PP(V),e}$ is a closed immersion.

\medskip\noindent
\begin{lem} \label{lem-equivdefn} 
The $1$-morphism $Y \rightarrow \Kbm{0,0}{\PP(V),e}$ is representable
by closed immersions and the image is the rank $2$ locus
$(\Kbm{0,0}{\PP(V),e})_2$ for $\phi_1:\mc{G}_1 \rightarrow \mc{F}_1$.
Moreover, for each $\phi_d$ the reduced substack of the rank $(d+1)$
locus $(\Kbm{0,0}{\PP(V),e})_{d+1}$ equals the image of $Y$.
\end{lem}

\medskip\noindent
\begin{proof}
It is clear that $Y \rightarrow \Kbm{0,0}{\PP(V),e}$ is injective on
geometric points; after all for a pair $([L],[f:C \rightarrow L])$, we
have that $L = f(C)$, so the line $[L]$ is uniquely determined by $f:C
\rightarrow \PP(V)$.  Moreover for a stable map $f:C\rightarrow X$,
the rank of $\phi_1$ restricted to the residue field of $[f]$ is at
least as big as $H^0(f(C),\OO_{\PP(V)}(1)|_{f(C)})$.  For a pure
$1$-dimensional subscheme of $\PP(V)$, this dimension is always at
least $2$, and equals $2$ only if $f(C)$ is a line.  Therefore every
geometric point of $(\Kbm{0,0}{\PP(V),e})_2$ is a geometric
point of $Y$.  The same sort of argument shows that for every $d$,
the rank $(d+1)$ locus of $\phi_d$ equals $Y$ (as sets of
geometric points).

\medskip\noindent
Moreover, since $(\Kbm{0,0}{\PP(V),e})_1$ is empty, on
$(\Kbm{0,0}{\PP(V),e})_2$ the quotient $\mc{E}$ of $\phi_1$ is a
locally free sheaf of rank $2$.  Then $\mc{E}$ is a quotient of
$H^0(\PP(V),\OO_{\PP(V)}(1))^\vee$ which induces a morphism from
$(\Kbm{0,0}{\PP(V),e})_2$ to $\mathbf{G}$, from which it is easy to
construct an inverse to $Y \rightarrow (\Kbm{0,0}{\PP(V),e})_2$.
\end{proof}

\medskip\noindent
Consider the restriction of $\phi_d$ to $Y$.  It is simpler to
phrase the results for the adjoint $\phi_d^\dagger$, but by (i)
of Lemma~\ref{lem-conn}, they are both equivalent.

\begin{lem} \label{lem-phi1} 
The kernel of $\phi_1^\dagger|_Y$ equals $\text{pr}_{\mathbb{G}}^*
T^\vee \subset V^\vee \otimes_\kappa \OO_Y$.  The cokernel of
$\phi_1^\dagger|_Y$ is a locally free sheaf $\mc{R}$ of rank $e-1$.
And the induced connecting map $\theta'_{\phi_1^\dagger,Y}:
\textit{Hom}_{\OO_Y} ( \mc{R}, \text{pr}_{\mathbb{G}}^* T^\vee )
\rightarrow \mc{J}/\mc{J}^2$ is an isomorphism of $\OO_Y$-modules.
\end{lem}

\medskip\noindent
\begin{proof}
The first two assertions follow from the proof of
Lemma~\ref{lem-equivdefn}; the details are left to the reader.  The
third assertion can probably be proved directly, but it also follows
from the deformation theory of Kontsevich stable maps developed in
\cite{B} and \cite{BF} (see also ~\cite[Sec. 3]{HS2}).  Since $Y$ is
smooth and since $\Kbm{0,0}{\PP(V),e}$ is smooth, the conormal sheaf
$\mc{J}/\mc{J}^2$ is a locally free $\OO_Y$-module.  The first step is
to describe this sheaf.

\medskip\noindent
As above, let $\pi:\mc{C} \rightarrow Y$ be the universal curve.  Let
$g:\mc{C} \rightarrow \PP(S)$ be the universal map (compatible with
projection to $\mathbb{G}$), and let $\text{pr}_{\PP(V)}:\PP(S)
\rightarrow \PP(V)$ be the obvious projection so that $f =
\text{pr}_{\PP(V)} \circ g$ is the universal map from $\mc{C}$ to
$\PP(V)$.  There is a perfect complex of amplitude $[-1,0]$ on
$\mc{C}$, denoted $L_f$, such that the object $(\mathbb{R}\pi_*
L_f^\vee)[1]$ in the derived category of $Y$ is quasi-isomorphic to
the restriction of the tangent bundle of $\Kbm{0,0}{\PP(V),e}$.
Similarly, there is a perfect complex of amplitude $[-1,0]$, denoted
$L_g$, such that the object $(\mathbb{R}\pi_* L_g^\vee)[1]$ in the
derived category of $Y$ is quasi-isomorphic to the vertical tangent
bundle of the morphism $\text{pr}_{\mathbb{G}}: Y \rightarrow
\mathbb{G}$.  These complexes are as follows:
\begin{equation}
\begin{CD}
& -1 & & 0 \\
L_f: & f^* \Omega^1_{\PP(V)} @> (df)^\dagger >> \Omega^1_{\mc{C}/Y} \\
L_g: & g^* \Omega^1_{\PP(S)/\mathbb{G}} @> (dg)^\dagger >>
\Omega^1_{\mc{C}/Y}
\end{CD}
\end{equation}
Of course the derivative of the morphism $\text{pr}_{\PP(V)}:\PP(S)
\rightarrow \PP(V)$ induces a surjective sheaf map from $f^*
\Omega^1_{\PP(V)}$ to $g^* \Omega^1_{ \PP(S)/\mathbb{G} }$ whose
kernel is just $g^*\OO_{\PP(S)}(-1)\otimes \text{pr}_{\mathbb{G}}^*
T^\vee$.  So there is a distinguished triangle
\begin{equation}
g^*\OO_{\PP(S)}(-1)\otimes \text{pr}_{\mathbb{G}}^* T^\vee [1]
\rightarrow L_f \rightarrow L_g
\end{equation}

\medskip\noindent
Applying the derived functor $\mathbb{R} \textit{Hom}_{ \OO_{\mc{C}} }
( * ,\OO_{\mc{C}} )$ to the distinguished triangle above produces a
distinguished triangle,
\begin{equation}
L_g^\vee \rightarrow L_f^\vee \rightarrow \textit{Hom}_{\OO_{\mc{C}}} \lt(
\text{pr}_{\mathbb{G}}^* T^\vee, g^*\OO_{\PP(S)}(1) \rt) [-1].
\end{equation}
Finally apply $\mathbb{R}\pi_*$ to this distinguished triangle to
get a distinguished triangle,
\begin{equation}
(\mathbb{R}\pi_* L_g^\vee)[1] \rightarrow (\mathbb{R}\pi_*
  L_f^\vee)[1] \rightarrow \textit{Hom}_{\OO_Y} \lt(
  \text{pr}_{\mathbb{G}}^* T^\vee, \mc{P}_1 \rt)[0],
\end{equation}
(obviously some minor details have been left out).
The derivative map from the vertical tangent bundle of
$\text{pr}_{\mathbb{G}}: Y \rightarrow \mathbb{G}$ to the restriction
of the tangent bundle of $\Kbm{0,0}{\PP(V),e}$ has cokernel isomorphic
to $\textit{Hom}_{\OO_Y} \lt( \text{pr}_{\mathbb{G}}^* T^\vee,
\mc{P}_1 \rt)$.

\medskip\noindent
The map $\phi_1^\dagger|_Y$ has image $\text{pr}_{\mathbb{G}}^*
S^\vee$.  Therefore inside of this cokernel there is a subsheaf
$\textit{Hom}_{\OO_Y} \lt( \text{pr}_{\mathbb{G}}^* T^\vee,
\text{pr}_{\mathbb{G}}^* S^\vee \rt)$.  This is just the pullback of
the tangent bundle of $\mathbb{G}$.  The cokernel of this subsheaf is
the normal bundle of $Y \rightarrow \Kbm{0,0}{\PP(V),e}$.  And, since
$\mc{R}$ is $\mc{P}_1/\text{pr}_{\mathbb{G}}^* S^\vee$ by definition,
this cokernel is canonically isomorphic to
$\textit{Hom}_{\OO_Y} \lt(\text{pr}_{\mathbb{G}}^* T^\vee, \mc{R}
\rt)$.  Therefore the dual sheaf, $\mc{J}/\mc{J}^2$, is canonically
isomorphic to $\textit{Hom}_{\OO_Y} \lt( \mc{R},
\text{pr}_{\mathbb{G}}^* T^\vee \rt)$.  Obviously this is very close
to what is needed.

\medskip\noindent
Using the canonical isomorphism above, the induced connecting map
$\theta'_{\phi_1^\dagger, Y}$ is an endomorphism of the locally
free sheaf $\textit{Hom}_{\OO_Y} \lt( \mc{R}, \text{pr}_{\mathbb{G}}^*
T^\vee \rt)$.  An endomorphism of a locally free sheaf is an
isomorphism iff the determinant of the endomorphism is invertible.
Since $Y$ is a proper, smooth, connected Deligne-Mumford stack, the
global sections of $\OO_Y$ are just the constants.  So to prove that
the determinant is invertible, it suffices to prove that it is
nonzero at a single point.

\medskip\noindent
The proof is reduced to a simple (slightly tedious) computation in local
coordinates.  Choose homogeneous coordinates $Y_0,\dots,Y_n$ on
$\PP(V)$, i.e. $Y_0,\dots,Y_n$ is an ordered basis for $V^\vee =
H^0(\PP(V),\OO_{\PP(V)}(1))$.  Choose homogeneous coordinates
$X_0,X_1$ on $\PP^1$.  Let $\AAA$ be the affine space associated to
the \emph{dual vector space} $W$ of linear transformations
\begin{equation}
W^\vee := \text{Hom}_{\kappa}\lt( 
H^0(\PP^1,\OO_{\PP^1}(e-2)), \text{span}\{Y_2,\dots,Y_n\} \rt).
\end{equation}
A basis for the vector space $W^\vee$, i.e. for the vector space of
linear forms on $\AAA$, is given by the tensors $(X_0^i
X_1^{e-2-i})^\vee \otimes Y_j$ for $j=2,\dots,n$ and $i=0,\dots,e-2$.
Define $F:\PP^1 \times \AAA \rightarrow \PP(V)$ to be the morphism
with $F^* \OO_{\PP(V)}(1) = \text{pr}_{\PP^1}^* \OO_{\PP^1}(e)$ and
where the pullback of homogeneous coordinates is defined by
\begin{equation}
\begin{array}{lll}
F^* Y_0 & = & \text{pr}_{\PP^1}^*X_0^e, \\
F^* Y_1 & = & \text{pr}_{\PP^1}^*X_1^e, \\
F^* Y_j & = & \sum_{i=0}^{e-2} \text{pr}_{\PP^1}^*
(X_0^{i+1}X_1^{e-1-i}) \cdot 
\text{pr}_{\AAA}^*( (X_0^i X_1^{e-2-i})^\vee \otimes Y_j), \
j=2,\dots,n
\end{array}
\end{equation}

\medskip\noindent
The morphism $F$ is a family of stable maps of degree $e$ over $\AAA$
and defines a $1$-morphism $\zeta:\AAA \rightarrow
\Kbm{0,0}{\PP(V),e}$.  The pullback by $\zeta$ of $\mc{P}_1$ is simply
the trivial vector bundle $H^0(\PP^1,\OO_{\PP^1}(e))$ on $\AAA$, and
the pullback of $\phi_1^\dagger$ is simply the map:
\begin{equation}
\begin{array}{lll}
\phi_1^\dagger(1\otimes Y_0) & = & 1\otimes X_0^e, \\
\phi_1^\dagger(1\otimes Y_1) & = & 1\otimes X_1^e, \\
\phi_1^\dagger(1\otimes Y_j) & = & \sum_{i=0}^{e-2} \lt( (X_0^i
X_1^{e-2-i})^\vee \otimes Y_j \rt) \otimes X_0^{i+1} X_1^{e-1-i}, \
j=2,\dots, n
\end{array}
\end{equation}
It follows that the rank $2$ locus is the origin $0\in \AAA$.  The
inverse image ideal sheaf $\zeta^{-1}\mc{J}$ is just the ideal of the
origin, i.e. the ideal with generators $(X_0^i X_1^{e-2-i})^\vee
\otimes Y_j$ for $i=0,\dots,e$ and $j=2,\dots,n$.  The kernel of
$\phi_1^\dagger|_0$ is $\text{span}\{ Y_2,\dots, Y_n \}$.  The image
of $\phi_1^\dagger|_0$ is $\text{span} \{ X_0^e,X_1^e \}$, and the
cokernel is $X_0X_1 \cdot H^0(\PP^1,\OO_{\PP^1}(e-2))$.  So the
pullback of $\textit{Hom}_{\OO_Y}(\mc{E}_Y,\mc{K}_Y)$ is the vector
space,
$$
\text{Hom}_\kappa \lt( H^0(\PP^1,\OO_{\PP^1}(e-2)),
\text{span}\{Y_2,\dots,Y_n \} \rt).
$$
Chasing through the snake diagram associated to $F$, the pullback of
the map $\theta'_{\phi_1^\dagger,Y}: \textit{Hom}_{\OO_Y}(\mc{E}_Y,
\mc{K}_Y) \rightarrow \mc{J}/\mc{J}^2$ is the map
\begin{equation}
(X_0^i X_1^{e-2-i})^\vee \otimes Y_j \mapsto (X_0^i X_1^{e-2-i})^\vee
  \otimes Y_j
\end{equation}
i.e. it is the identity map.  This proves that
$\theta_{\phi_1^\dagger,Y}$ is an isomorphism when restricted to the
image of $0\in \AAA$.  As mentioned above, this suffices to prove that
$\theta_{\phi_1^\dagger,Y}$ is everywhere an isomorphism.  
\end{proof}

\begin{rmk} \label{rmk-thetaid}
In fact,
via the canonical isomorphism of $\mc{J}/\mc{J}^2$ with
$\textit{Hom}_{\OO_Y}(\mc{E}_Y,\mc{K}_Y)$ induced by the deformation
theory computation, the endomorphism $\theta_{\phi_1^\dagger,Y}$ is
the identity map.  Since this is not used here, it is not proved.
\end{rmk}

\begin{notat} \label{notat-Rd}
For every nonnegative integer $d$, denote by $\mc{A}_d$ the locally
free $\OO_{Y}$-module, $\text{pr}_{\mathbb{G}}^*(
\text{Sym}^{d-1}(S^\vee) )^\vee$, denote by
$\mc{R}_d$ the locally free
$\OO_Y$-module, $\mc{R} \otimes_{\OO_Y} \text{pr}_{\mathbb{G}}^*
\text{Sym}^{d}(S^\vee)$, and denote by $T^\vee_d$ the locally free
$\OO_{\mathbb{G}}$-module,   
$T^\vee \otimes_{\OO_{\mathbb{G}}} 
\text{Sym}^{d}(S^\vee).$
\end{notat}

\begin{prop} \label{prop-phid} 
Using the isomorphisms from Lemma~\ref{lem-phi1}, the following hold.
\begin{enumerate}
\item[(i)]
For each $d\geq 1$, the kernel of $\phi_d^\dagger|_Y$ is
$\text{pr}_{\mathbb{G}}^* F^1 \subset \text{Sym}^d(V^\vee)
\otimes_\kappa \OO_Y$.
\item[(ii)] 
The cokernel of $\phi_d^\dagger|_Y$ is canonically isomorphic to
$\mc{R}_{d-1}$.
\item[(iii)]
The induced connecting map
\begin{equation}
\theta'_{\phi_d^\dagger,Y}: \textit{Hom}_{\OO_Y} (\mc{R}_{d-1}, 
\text{pr}_{\mathbb{G}}^* F^1 ) 
\rightarrow \mc{J}/\mc{J}^2
\end{equation}
is the zero map on the subsheaf $\textit{Hom}_{\OO_Y} (\mc{R}_{d-1},
\text{pr}_{\mathbb{G}}^* F^2 )$.  
\item[(iv)]
Identify $F^1/F^2$ with $T^\vee \otimes_{\OO_{\mathbb{G}}}
\text{Sym}^{d-1}(S^\vee)$, i.e. $T^\vee_{d-1}$,
and identify $\mc{J}/\mc{J}^2$ with
$\textit{Hom}_{\OO_Y}\lt( \text{pr}_{\mathbb{G}}^* T^\vee, \mc{R}
\rt)$ using $\theta'_{\phi_1^\dagger,Y}$.  The following map,
\begin{equation}
\begin{array}{r}
\theta_{\phi_d^\dagger,Y}'': \textit{Hom}_{\OO_Y} ( \mc{R}_{d-1}, 
\text{pr}_{\mathbb{G}}^* T^\vee_{d-1})
\rightarrow \textit{Hom}_{\OO_Y} ( \mc{R},
\text{pr}_{\mathbb{G}}^* T^\vee ),
\end{array}
\end{equation}
induced by $\theta_{\phi_d^\dagger,Y}'$,
equals the map obtained by contracting the
$\text{Sym}^{d-1}(S^\vee)$ factors.
\end{enumerate}
\end{prop}

\begin{proof}
\textbf{(i)}:
The sheaf $\mc{P}_d$ is defined to be $\pi_* f^* \OO_{\PP(V)}(d)$.
Consider the fiber product $\PP(S) \times_{\mathbb{G}} Y$.  There is
an induced finite, flat, surjective
morphism $(g,\pi):\mc{C} \rightarrow \PP(S) \times_{\mathbb{G}}
Y$.  And $f^* \OO_{\PP(V)}(d)$ equals $(g,\pi)^* \text{pr}_{\PP(S)}^*
\OO_{\PP(S)}(d)$.  Since $\pi = \text{pr}_{Y} \circ (g,\pi)$, 
\begin{equation}
\mc{P}_d = (\text{pr}_{Y})_* (g,\pi)_* (g,\pi)^*
\text{pr}_{\PP(S)}^* \OO_{\PP(S)}(d).
\end{equation}
There is a canonical sheaf map $\text{pr}_{\PP(S)}^* \OO_{\PP(S)}(d)
\rightarrow (g,\pi)_* (g,\pi)^* \text{pr}_{\PP(S)}^* \OO_{\PP(S)}(d)$.
Because $(g,\pi)$ is faithfully flat, this sheaf map is
injective.  Since pushforward is left exact, there is an injective
sheaf map from $(\text{pr}_{Y})_* \text{pr}_{\PP(S)}^*
\OO_{\PP(S)}(d)$ to $\mc{P}_d$.  But of course the first sheaf is just
$\text{pr}_{\mathbb{G}}^* \text{Sym}^d(S^\vee)$.  And the injective
sheaf map $\text{Sym}^d(S^\vee) \rightarrow \mc{P}_d$ is the image of
$\phi_d^\dagger|_Y$.  Therefore the kernel of $\phi_d^\dagger|_Y$ is
the pullback of the kernel of $\text{Sym}^d(V^\vee) \otimes_\kappa
\OO_{\mathbb{G}} \rightarrow \text{Sym}^d(S^\vee)$, i.e. the first
filtered subsheaf $F^1$ of $\text{Sym}^d(V^\vee) \otimes_\kappa
\OO_{\mathbb{G}}$.  This proves that the kernel of $\phi_d^\dagger|_Y$
equals $\text{pr}_{\mathbb{G}}^* F^1$.  

\medskip\noindent
\textbf{(ii)}:
On $\PP(S)$ there is a short exact sequence,
\begin{equation}
0 \rightarrow \text{pr}_{\mathbb{G}}^*
\mathbb{S}_{(d-1,1)}(S^\vee)\otimes \OO_{\PP(S)}(-1)
\rightarrow \text{pr}_{\mathbb{G}}^* \text{Sym}^{d-1}(S^\vee)
\rightarrow \OO_{\PP(S)}(d-1) \rightarrow 0,
\end{equation}
where $\mathbb{S}_{(d-1,1)}$ is the \emph{Schur functor} as defined in
~\cite[Sec. 6.1]{FH}.  Twist this sequence by $\OO_{\PP(S)}(1)$ and
pullback by $(g,\pi)$ to get a short exact sequence of
$\OO_{\mc{C}}$-modules.  Pushing forward by $\pi$ yields a long exact
sequence of higher direct image sheaves.  Since $\pi:\mc{C}
\rightarrow Y$ is a flat family of at-worst-nodal curves of genus $0$,
$R^1\pi_* \OO_{\mc{C}}$ is zero.  Therefore, the long
exact sequence reduces to the following short exact sequence,
\begin{equation}
0 \rightarrow \text{pr}_{\mathbb{G}}^* \mathbb{S}_{(d-1,1)}(S^\vee)
\rightarrow \text{pr}_{\mathbb{G}}^* \text{Sym}^{d-1}(S^\vee) \otimes
\mc{P}_1 \rightarrow \mc{P}_d \rightarrow 0.
\end{equation}
Of course, there is a commutative diagram with exact rows:
\begin{equation}
\begin{array}{ccccccccc}
0 & \rightarrow & \text{pr}_{\mathbb{G}}^*
\mathbb{S}_{(d-1,1)}(S^\vee) & \rightarrow &
\text{pr}_{\mathbb{G}}^* \text{Sym}^{d-1}(S^\vee) \otimes
\text{pr}_{\mathbb{G}}^* S^\vee & \rightarrow &
\text{pr}_{\mathbb{G}}^* \text{Sym}^d(S^\vee) & \rightarrow & 0 \\
 & & \text{Id} \downarrow & & \downarrow \text{Id}\otimes
\phi_1^\dagger & & \downarrow \phi_d^\dagger \\
0 & \rightarrow & \text{pr}_{\mathbb{G}}^* \mathbb{S}_{(d-1,1)}(S^\vee)
& \rightarrow & \text{pr}_{\mathbb{G}}^* \text{Sym}^{d-1}(S^\vee) \otimes
\mc{P}_1 & \rightarrow & \mc{P}_d & \rightarrow & 0
\end{array}
\end{equation}
Applying the snake lemma to this diagram yields an isomorphism of the
cokernel of $\phi_d^\dagger$ with $\mc{R}\otimes
\text{pr}_{\mathbb{G}}^* \text{Sym}^{d-1}(S^\vee)$, i.e. $\mc{R}_{d-1}$.  

\medskip\noindent
\textbf{(iii)}
In the special case $d=1$, (iii) follows from Lemma~\ref{lem-phi1}.  
Thus suppose that $d > 1$.  The proof of (iii) uses the fact
that there is a commutative diagram of sheaves on
$\Kbm{0,0}{\PP(V),e}$:
\begin{equation}
\begin{CD}
V^\vee \otimes_K \text{Sym}^{d-1}(V^\vee) \otimes_K \OO @>
\phi_1^\dagger \otimes \text{Id} >> \mc{P}_1 \otimes_K
\text{Sym}^{d-1}(V^\vee) \\
@V \psi_{\mc{G}} VV @VV \psi_{\mc{F}} V \\
\text{Sym}^d(V^\vee) \otimes_K \OO @> \phi_d^\dagger >> \mc{P}_d
\end{CD}
\end{equation}
Associated to $\phi_1^\dagger \otimes \text{Id}$ is the induced
connecting map,
\begin{equation}
\theta'_{\phi_1^\dagger \otimes \text{Id},Y}:
\textit{Hom}_{\OO_Y} ( \mc{R} \otimes_K \text{Sym}^{d-1}(V^\vee),
\text{pr}_{\mathbb{G}}^* T^\vee \otimes_K
\text{Sym}^{d-1}(V^\vee) )  \rightarrow \mc{J}/\mc{J}^2.
\end{equation}
Of course this is obtained from $\theta'_{\phi_1^\dagger,Y}$ by
contracting the $\text{Sym}^{d-1}(V^\vee)$ factors.
Define $\theta''_{\phi_1^\dagger \otimes \text{Id}, Y}$ to be the
restriction of $\theta'_{\phi_1^\dagger \otimes \text{Id}, Y}$ to the
subsheaf 
$
\textit{Hom}_{\OO_Y} ( \mc{R}_{d-1},
\text{pr}_{\mathbb{G}}^* T^\vee \otimes_K
\text{Sym}^{d-1}(V^\vee) ).
$
By (iv) of Lemma~\ref{lem-conn}, there is a commutative diagram
of induced connecting maps,
\begin{equation}
\begin{CD}
\textit{Hom}_{\OO_Y} ( \mc{R}_{d-1},
\text{pr}_{\mathbb{G}}^* T^\vee \otimes_K
\text{Sym}^{d-1}(V^\vee) ) @> \theta'_{\phi_1^\dagger\otimes
  \text{Id},Y} >> \mc{J}/\mc{J}^2 \\
@V\psi VV @VV \text{Id} V \\
\textit{Hom}_{\OO_Y} ( \mc{R}_{d-1} 
\text{pr}_{\mathbb{G}}^* F^1 ) @> \theta'_{\phi_d^\dagger, Y} >>
\mc{J}/\mc{J}^2 
\end{CD}
\end{equation}
Since $\theta'_{\psi_1^\dagger\otimes \text{Id},Y}$ is obtained by
contracting the $\text{Sym}^{d-1}(V^\vee)$ factors,
in particular the kernel of
$\theta_{\phi_1^\dagger \otimes \text{Id}, Y}$ contains the 
subsheaf
$
\textit{Hom}_{\OO_Y} ( \mc{R}_{d-1}, 
\text{pr}_{\mathbb{G}}^* T^\vee \otimes_{\OO_Y} F^1 ).
$
Therefore the kernel of $\theta'_{\phi_d^\dagger,Y}$ contains the
image under $\psi$ of this subsheaf.  But the image under $\psi$ is
just
$
\textit{Hom}_{\OO_Y} (\mc{R}_{d-1}, 
\text{pr}_{\mathbb{G}}^*F^2 ).
$
Forming the quotient by this sheaf gives another commutative
diagram,
\begin{equation}
\begin{CD}
\textit{Hom}_{\OO_Y} ( \mc{R}_{d-1}, 
\text{pr}_{\mathbb{G}}^* T^\vee \otimes_K
\text{Sym}^{d-1}(V^\vee)  ) 
@> \theta'_{\phi_1^\dagger\otimes
  \text{Id},Y} >> \mc{J}/\mc{J}^2 \\
@V\psi'' VV @VV \text{Id} V \\
\textit{Hom}_{\OO_Y} ( \mc{R}_{d-1}, \text{pr}_{\mathbb{G}}^* T_{d-1})
@> \theta''_{\phi_d^\dagger, Y} >>
\mc{J}/\mc{J}^2 
\end{CD}
\end{equation}
Since $\psi''$ is surjective, $\theta''_{\phi_d^\dagger,Y}$ is the
unique morphism making the above diagram commute.  But using the fact
that $\theta'_{\phi_1^\dagger \otimes \text{Id}, Y}$ is obtained from
contracting the $\text{Sym}^{d-1}(V^\vee)$ factors, it is clear that
the diagram also commutes when 
$\theta''_{\phi_d^\dagger,Y}$ is replaced by the map obtained from
$\theta'_{\phi_1^\dagger,Y}$ by contracting the
$\text{Sym}^{d-1}(S^\vee)$ factors.  Therefore
$\theta''_{\phi_d^\dagger,Y}$ equals the map obtained from
$\theta'_{\phi_1^\dagger,Y}$ by contracting the
$\text{Sym}^{d-1}(S^\vee)$ factors.  
\end{proof}

\medskip\noindent
\begin{cor} \label{cor-phid} 
Regarding the restriction $\phi_d|_Y:\mc{G}_d|_Y \rightarrow
\mc{F}_d|_Y$, the following hold.
\begin{enumerate}
\item[(i)] 
The cokernel $\mc{E}_Y$ is canonically isomorphic to
$\text{pr}_{\mathbb{G}}^* (F^1)^\vee$.
\item[(ii)] 
The kernel $\mc{K}_Y$ is canonically isomorphic to $(\mc{R}_{d-1})^\vee$.
\item[(iii)] 
The induced connecting map $\theta'_{\phi_d,Y}$ is canonically
isomorphic to the induced connecting map $\theta'_{\phi_d^\dagger,Y}$.
\end{enumerate}
\end{cor}

\medskip\noindent
\begin{proof}
This follows from (i) of Lemma~\ref{lem-conn} and
Proposition~\ref{prop-phid}.
\end{proof}

\section{Proof of the main theorem} \label{sec-proof} 

\medskip\noindent 
Fix an integer $e \geq 1$.  

\begin{notat} \label{notat-B}
To simplify notation, in this
section denote $B=\Kbm{0,0}{\PP(V),e}$.  
For each integer $d$, denote by $\pi:C_d \rightarrow B$ the projective
Abelian cone of the 
coherent $\OO_B$-module, $\mc{E}_d = \text{Coker}(\phi_d)$.  When
there is no risk of confusion, $C_d$ is denoted by $C$.
\end{notat}

By Lemma~\ref{lem-PE}, $C_d$
is the
Deligne-Mumford stack parametrizing pairs $([X],[f:C \rightarrow X])$
of a hypersurface of degree $d$, $X\subset \PP(V)$, 
together with a stable map,
$[f:C\rightarrow X]$, in $\Kbm{0,0}{X,e}$.  In this section
the singularities of the cone $C_d$ are described.  The reader is
reminded that $K$ is an algebraically closed field of 
characteristic $0$. 

\medskip\noindent
The simplest case is $e=1$.  The next two results are
already known, in fact in arbitrary characteristic
~\cite[Thm. V.4.3]{K}.  Only that part used here is proved (or rather
reproved).  

\medskip\noindent
\begin{prop} \label{prop-e=1} 
If $e=1$, then $B$ is a smooth projective scheme and for all $d\geq 1$
the morphism $\pi_d:C_d \rightarrow B$ is a projective bundle of the
expected dimension.  In particular $C_d$ is a geometrically
irreducible, smooth scheme of the expected dimension.
\end{prop}

\begin{proof}
Since $e=1$, $Y$ equals $B$, which is simply the Grassmannian
$\mathbb{G}$.  Thus $\mc{R}$ is the zero sheaf.  And by
Corollary~\ref{cor-phid}, the cokernel $\mc{E}_d$ is locally free of
the expected dimension.  Therefore $\pi:C \rightarrow B$ is a
projective bundle.
\end{proof}

\begin{thm} \label{thm-e=1} \cite[V.4.3]{K}
If $e=1$ and if $d > 2n-3$, then the projection morphism $h_d:C_d
\rightarrow \PP H^0\lt( \PP(V), \OO_{\PP(V)}(d) \rt)$ is not
surjective, i.e. the general fiber is empty.  If $d \leq 2n-3$, then
the projection morphism $h_d$ is surjective, and the general fiber is
a smooth scheme of the expected dimension $2n-d-3$.  Moreover if $d <
2n-3$ and $(d,n) \neq (2,3)$, then the general fiber is geometrically
connected.
\end{thm}

\medskip\noindent
\begin{proof}
For the full proof, the reader should consult ~\cite{K}.  
Only that part of the theorem used later is proved here,
namely that the
general fiber is nonempty and smooth for $d \leq n-1$ and that the
general fiber is geometrically connected for $d \leq n-2$.

\medskip\noindent
Suppose that $d\leq n-1$.  Since $C_d$ is irreducible of the expected
dimension, $h_d$ is surjective iff the general fiber has the expected
dimension.  To prove the general fiber has the expected dimension, it
suffices to find one pair $([X],[L])$ consisting of a hypersurface
$X\subset \PP(V)$ and a line $L\subset X$ such that $X$ is smooth
along $L$ and such that $H^1(L,N_{L/X})$ is zero.  Then the Zariski
tangent space of the fiber, i.e. $H^0(L,N_{L/X})$, has the expected
dimension proving that on a nonempty (hence dense) open subset of
$C$, $h_d$ has the expected fiber dimension.  Choose homogeneous
coordinates $Y_0,Y_1,Y_2,\dots,Y_n$ on $\PP(V)$.  Define $L$ to be the
vanishing locus of $Y_2,\dots,Y_n$.  Define $X\subset \PP(V)$ to be
the hypersurface with defining equation
\begin{equation}
 F:= \sum_{j=2}^{d+1} Y_0^{d+1-j}Y_1^{j-2} Y_j
\end{equation}
At every point of $L$, either the partial derivative $F_2$ is nonzero
or the partial derivative $F_{d+1}$ is nonzero.  Therefore $X$ is
smooth along $L$.  Moreover, by the usual exact sequence,
\begin{equation}
0 \rightarrow N_{L/X} \rightarrow \OO_L(1)^{n-1}
\xrightarrow{F_2,\dots,F_{d+1},0,\dots,0} \OO_L(d) \rightarrow 0,
\end{equation}
$N_{L/X}$ is isomorphic to $\OO_L^{d-1}\oplus \OO_L(1)^{n-d-1}$.
Therefore $H^1(L,N_{L/X})$ is zero proving that $h_d$ is
surjective and the generic fiber has the expected dimension.  Since
$C_d$ is smooth and since $\PP H^0(\PP(V),\OO_{\PP(V)}(d) )$ is
smooth, it follows by generic smoothness that the generic fiber of
$h_d$ is everywhere smooth.

\medskip\noindent
Next it is proved that the generic fiber of $h_d$ is connected if $d
\leq n-2$.  In this case, by the same sort of dimension computation as
above, for a general hypersurface $X\subset \PP(V)$,
every irreducible component of $\Kbm{0,1}{X,1}$ surjects to $X$ under
the evaluation morphism $\text{ev}:\Kbm{0,1}{X,1} \rightarrow X$.
Therefore, to prove that $\Kbm{0,1}{X,1}$ is connected, and thus that
$\Kbm{0,0}{X,1}$ is connected, it suffices to prove that every fiber
of $\text{ev}$ is connected.  Now for a given $p\in X$, the set of
lines in $\PP(V)$ containing $p$ is canonically isomorphic to the
projective space $\PP(V/L_p)$, where $L_p \subset V$ is the
one-dimensional vector subspace corresponding to $p$.  Choose
homogeneous coordinates so that $p=[1:0:\dots:0]$.  Then the Taylor
expansion of the defining equation $F$ of $X$ about the point $p$ has
the form,
\begin{equation}
F = Y_0^{d-1}F_1(Y_1,\dots, Y_n) + Y_0^{d-2}F_2(Y_1,\dots,Y_n) +
\dots + Y_0F_{d-1}(Y_1,\dots, Y_n) + F_d(Y_1,\dots, Y_n),
\end{equation}
where each $F_i$ is a homogeneous polynomial of degree $i$ in
$Y_1,\dots,Y_n$.  A line passing through $p$ with parametric equation
$[(1-t):tY_1:tY_2:\dots:tY_n]$ is contained in $X$ iff the equations
$F_1(Y_1,\dots,Y_n), \dots , F_d(Y_1,\dots, Y_n)$ are all zero.  It
should be observed that some of the homogeneous equations $F_i$ may be
identically zero.  Nonetheless, the common zero locus of $F_1,\dots,
F_d$ in $\PP(V/L_p)$ is an intersection of at most $d$ hypersurfaces
in a projective space of dimension $n-1$.  In a projective space of
dimension $n-1$, an intersection of at most $n-2$ hypersurfaces is
always connected.  Since $d \leq n-2$, the common
zero locus of $F_1,\dots, F_d$ is connected.  Therefore every fiber of
$\text{ev}$ is connected, proving that the fiber $\Kbm{0,0}{X,1}
= h_d^{-1}([X])$ is connected.
\end{proof}

\medskip\noindent
Now suppose that $e>1$.  Consider the closed immersion $Y
\hookrightarrow B$.  

\begin{notat} \label{notat-N}
Because the conormal sheaf of $Y$ is locally
free, the normal cone $C$ is denoted by the letter $N$ (also to avoid
confusion with the cone $C_d$ which we are studying).  
\end{notat}

\medskip\noindent
Associated to
the closed immersion $Y \rightarrow B$, there is the deformation to the
normal cone $(\varrho:M \rightarrow \PP^1, \iota:Y \times \PP^1
\rightarrow M, B_Y \rightarrow M, \PP(N\oplus \mathbf{1}) \rightarrow
M)$ as in Definition~\ref{defn-dnc}.  Define $M^o = M - B_Y$.  In
particular, the intersection of $M^o$ with $\PP(N \oplus \mathbf{1})$
is just $N$.

\medskip\noindent
Let $\widetilde{\phi}_d:\widetilde{\mc{G}} \rightarrow \mc{F}_M$ be
the elementary-transform-up of the pullback of $\phi_d$ as described
in Lemma~\ref{lem-elemtrans}.  Let $\widetilde{\mc{E}}_d$ be the
cokernel of $\widetilde{\phi}_d|_{M^o}$ and let
$\widetilde{\pi}_d:\widetilde{C}_d \rightarrow M^o$ be the projective
Abelian cone parametrizing rank $1$ locally free quotients of
$\widetilde{\mc{E}}_d$.  Observe that over the open subset
$\varrho^{-1}(\AAA^1) = B \times \AAA^1$, $\widetilde{\mc{E}}_d$ is
simply $\text{pr}_B^* \mc{E}_d$ and $\widetilde{C}_d$ is simply $C_d
\times \AAA^1$.

\medskip\noindent
\begin{lem} \label{lem-restrict} 
Let $e \geq 2$.  If $d+e \leq n$, then the fiber product
$\widetilde{C}_d \times_{M^o} N$ is an integral, normal, local
complete intersection scheme of the expected dimension having at worst
canonical singularities.
\end{lem}

\medskip\noindent
\begin{proof}
By Lemma~\ref{lem-phi1}, the normal bundle $N\rightarrow Y$ is the
vector bundle associated to the locally free sheaf
$\textit{Hom}_{\OO_Y}(\mc{R}^\vee, \text{pr}_{\mathbb{G}}^* T)$,
i.e. $N = M^{(0)}(Y,\mc{R}^\vee, \text{pr}_{\mathbb{G}}^* T)$ in the
notation of Section~\ref{sec-deter}.  Denote by
$\mc{A}_{d-1}$ the locally free sheaf $\text{pr}_{\mathbb{G}}^* \lt(
\text{Sym}^{d-1}(S^\vee) \rt)^\vee$, as in Notation~\ref{notat-Rd}.  
This is a locally free sheaf of
rank $d$ on $Y$.

\medskip\noindent
By Corollary~\ref{cor-phid}, the kernel of $\phi_d|_Y$ is the locally
free sheaf
\begin{equation}
\mc{K}_Y = 
\mc{A}_{d-1} \otimes_{\OO_Y} \lt(\mc{R}^\vee \rt) 
\end{equation}
and the cokernel $\mc{E}_Y$ of $\phi_d|_Y$ fits into a short exact
sequence of locally free sheaves
\begin{equation}
0 \rightarrow \mc{A}_{d-1} \otimes_{\OO_Y} \lt( \text{pr}_{\mathbb{G}}^*
T\rt) 
\rightarrow \mc{E}_Y \rightarrow \text{pr}_{\mathbb{G}}^*(F^2)^\vee
\rightarrow 0.
\end{equation}
Denote the first sheaf in this sequence by $\mc{E}'_Y$ and denote the
third sheaf in this sequence by $\mc{A}'$.  By
Notation~\ref{notat-gamma}, the cokernel of $\widetilde{\phi}_d$ on
the Cartier divisor $N\subset M^o$ equals the cokernel of the sheaf
map $\gamma: \pi_Y^* \mc{K}_Y \rightarrow \pi_Y^* \mc{E}_Y$.  This uses
the fact that $\OO_{\PP(N\oplus \mathbf{1})}(1)$ is
canonically trivialized on $N$, so that all ``twists'' by this
sheaf are canonically just ``twists'' by the structure sheaf.  
By Lemma~\ref{lem-thetagamma},
the map $\gamma$ is the unique map induced by $\theta_{\phi_d,Y}$.  By
(i) of Lemma~\ref{lem-conn}, this means that $\gamma$ is the
transpose of the unique map induced by $\theta_{\phi_d^\dagger,Y}$.
By (iii) of Lemma~\ref{prop-phid}, the map
$\theta_{\phi_d^\dagger,Y}$ is the map induced by the universal
homomorphism from $\text{pr}_{\mathbb{G}}^*(T^\vee)$ to $\mc{R}$.

\medskip\noindent
Putting all the pieces together, there are two conclusions:  First, the
image of $\gamma$ is actually the image of a sheaf map,
\begin{equation}
\gamma': \mc{K}_Y \otimes_{\OO_Y} \OO_N \rightarrow \mc{E}'_Y
\otimes_{\OO_Y} \OO_N,
\end{equation}
so that there is a short exact sequence (which is split Zariski locally
over $Y$),
\begin{equation}
0 \rightarrow \text{Coker}(\gamma') \rightarrow \text{Coker}(\gamma)
\rightarrow \text{pr}_Y^* \mc{A}' \rightarrow 0.
\end{equation}
And second, denoting by $\psi: \mc{R}^\vee \otimes_{\OO_Y} \OO_N
\rightarrow \text{pr}_{\mathbb{G}}^* T \otimes_{\OO_Y} \OO_N$ the
universal sheaf homomorphism on $N$, the sheaf map $\gamma'$ is
just $\text{Id} \otimes \psi$,
\begin{equation}
\gamma' = \text{Id} \otimes \psi:  
\mc{A}_{d-1} \otimes_{\OO_Y} \lt(\mc{R}^\vee \rt) \otimes_{\OO_Y} \OO_N
\rightarrow 
\mc{A}_{d-1} \otimes_{\OO_Y} \lt( \text{pr}_{\mathbb{G}}^* T\rt)
\otimes_{\OO_Y} \OO_N. 
\end{equation}

\medskip\noindent
Now the rank of $\mc{R}^\vee$ is $e-1$, the rank of
$\text{pr}_{\mathbb{G}}^* T$ is $n-1$ and the rank of $\mc{A}_{d-1}$ is $d$.
By assumption, $d \leq (n-1) - (e-1)$.  This means that,
Zariski locally over $Y$, the hypotheses of
Proposition~\ref{prop-cone} hold.  So $\widetilde{C}_d
\times_{M^o} N$ is a normal, integral, local complete intersection
scheme of the expected dimension having at worst canonical singularities.
\end{proof}

\begin{notat} \label{notat-W}
Denote by $W \subset B$ the maximal open substack over which $C_d$ is
an integral, normal scheme of the expected dimension having at worst
canonical singularities. Observe that $W\times \AAA^1 \subset B\times
\AAA^1$ is the maximal open substack over which $\widetilde{C}_d
\times_{M^o} (B\times \AAA^1)$ is an integral, normal scheme of the
expected dimension having at worst canonical singularities.  Denote by $W'
\subset M^o$ the maximal open substack over which
$\widetilde{C}_d$ is an integral, normal scheme of the expected
dimension having at worst canonical singularities.
\end{notat}

\medskip\noindent
\begin{lem} \label{lem-VcontY} 
If $e \geq 2$ and $d+e \leq n$, then the open substack $W \subset B$
contains the closed substack $Y \subset B$.
\end{lem}

\medskip\noindent
\begin{proof}
By Lemma~\ref{lem-restrict}, by Corollary~\ref{cor-discrep}, by
Proposition~\ref{prop-relcan}, and by (iii) of
Corollary~\ref{cor-spec} applied to $N \subset M^o$, the open substack
$W'$ contains $N$.

\medskip\noindent
There is one slight finesse in checking the hypothesis of (iii) of
Corollary~\ref{cor-spec}:  In case $n-e \geq 2$, 
$\text{codim}_N(N_{g-1}) = (f-(g-1))(g-(g-1)) = f-g+1$ equals $(n-1) -
(e-1) + 1 = n-e+1 \geq 3$. Therefore Lemma~\ref{lem-D} proves that the
hypothesis of (iii) of Corollary~\ref{cor-spec} is satisfied.

\medskip\noindent
The one remaining case is $d=1, e=n-1$.  In this case, 
there is an \emph{ad hoc} argument.  The morphism
$\widetilde{\phi}_1$ restricted to $N$ is just the universal sheaf map
$\psi$.  In this special case $\mc{D}_{\phi_N}$ sits inside $N
\times_Y \PP(\text{pr}_{\mathbb{G}}^* T)$.  The projection
$\mc{D}_{\phi_N} \rightarrow \PP(\text{pr}_{\mathbb{G}}^* T)$ is a
Zariski locally trivial bundle.  Given a closed point $p\in Y$ and a
one-dimensional subspace $L\subset \text{pr}_{\mathbb{G}}^* T|_p$, the
fiber over this point, considered as a subvariety of $N =
M^{(0)}(Y,\mc{R}^\vee, \text{pr}_{\mathbb{G}}^* T)$, equals the cone
whose vertex set is $Hom_{\kappa(p)}(\mc{R}^\vee|_p, L)$ and whose
quotient by the vertex set is the set of non-invertible 
linear maps in
$Hom_{\kappa(p)}(\mc{R}^\vee|_p, \text{pr}_{\mathbb{G}}^* T|_p/L).$
Observe that this second vector space is
essentially just the vector space of square $(e-1)\times(e-1)$
matrices.  In the special case $e=2$, the cone is just a linear space
and so it is smooth.  In case $e\geq 3$ the vertex set has codimension
$(e-1)^2 -1 \geq 3$ in the fiber of $\mc{D}_{\phi_N}$, and the
singular locus of the quotient has codimension $4-1 = 3$:
$Hom_{\kappa(p)}(\mc{R}^\vee|_p, \text{pr}_{\mathbb{G}}^*
T|_p/L)_{e-3}$ has codimension $4$.  Therefore the singular locus has
codimension $3$ in the fiber of $\mc{D}_{\phi_N}$.  So the fiber is
normal, which implies that $\mc{D}_{\phi_N}$ is normal.  Therefore
when $d=1$ and $e=n-1$, the hypotheses of (iii) of
Corollary~\ref{cor-spec} are again satisfied.

\medskip\noindent
Of course $W' \cap \varrho^{-1}(\AAA^1)$ equals $W \times \AAA^1$.
Let $p\in Y$ be any point and consider $\iota(\{p\} \times \PP^1)
\subset M$.  By construction of the deformation to the normal cone
from Definition~\ref{defn-dnc}, $\iota(p,\infty)$ is the point on the
zero section of $N\rightarrow Y$ over $p\in Y$.  In particular
$\iota(\{p\} \times \PP^1) \subset M^o$ and $\iota(p,\infty) \in N$.
Therefore $\iota(\{ p \} \times \PP^1)$ intersects $W'$.  So $\iota(\{
p \} \times \AAA^1)$ intersects $W' \cap \varrho^{-1}(\AAA^1)$, i.e
$\{ p \} \times \AAA^1$ intersects $W\times \AAA^1$.  Therefore $p\in
W$.  Therefore $Y$ is contained in $W$.
\end{proof}

\begin{thm} \label{thm-main} 
If $e\geq 2$ and if $d+e \leq n$, then $C_d$ is an integral, normal,
local complete intersection stack of the expected dimension having at
worst 
canonical singularities.
\end{thm}

\begin{proof}
By Lemma~\ref{lem-VcontY}, the open substack $W$ contains $Y$.  Now
the automorphism group $\text{GL}(V)$ acts on $\PP(V)$ and thus on
$B$.  Moreover the sheaves $\mc{G}_d$ and $\mc{F}_d$ have natural
$\text{GL}(V)$-linearizations and the morphism $\phi_d$ is
$\text{GL}(V)$-equivariant.  Therefore $W$ is a
$\text{GL}(V)$-invariant open substack of $B$.  So to prove that $W =
B$, it suffices to prove that the closure of every
$\text{GL}(V)$-orbit intersects $Y$.

\medskip\noindent
Let $f:D \rightarrow \PP(V)$ be any stable map of genus $0$ and degree
$e$.  Choose a direct sum decomposition $V = V_2 \oplus V_{n-1}$ so
that $\PP(V_{n-1}) \subset \PP(V)$ is disjoint from $f(D)$.  Consider
the $\mathbb{G}_m$-action $m: \mathbb{G}_m \times \PP(V) \rightarrow
\PP(V)$ given by $t\cdot(v,v') = (v,tv')$, where $v\in V_2, v'\in
V_{n-1}$.  This defines an action of a subgroup scheme of
$\text{GL}(V)$.  Acting on $[f:D \rightarrow \PP(V)]$ yields a
$1$-morphism $\zeta:\mathbb{G}_m \rightarrow B$.  The limit as
$t\rightarrow 0$ of this action is simply the stable map $g\circ f: D
\rightarrow \PP(V_2) \subset \PP(V)$ where $g:\PP(V) - \PP(V_{n-1})
\rightarrow \PP(V_2)$ is the projection map.  In particular, $g\circ
f$ is a multiple cover of the line $\PP(V_2)$.  Therefore the closure
of the $\text{GL}(V)$-orbit of $[f:D\rightarrow \PP(V)]$ intersects
$Y$ in the point $[g\circ f:D \rightarrow \PP(V)]$.  It follows that
$W$ is all of $B$, i.e. $C_d$ is an integral, normal, local complete
intersection stack of the expected dimension having at worst canonical
singularities.
\end{proof}

\begin{cor} \label{cor-main} 
If $e \geq 2$ and if $d+e \leq n$, then for a general hypersurface
$X\subset \PP(V)$ of degree $d$, the Kontsevich moduli space
$\Kbm{0,0}{X,e}$ is an integral, normal, local complete intersection
stack of the expected dimension $(n+1-d)e+(n-3)$ having at worst canonical
singularities.
\end{cor}

\begin{proof}
By Theorem~\ref{thm-main}, $C_d$ is integral, normal, Gorenstein and
canonical.  Consider the projection $h_d:C_d \rightarrow \PP
H^0(\PP(V), \OO_{\PP(V)}(d) )$.  The pullback of the hyperplane linear
system gives a base-point-free linear system on $C_d$.  By repeated
application of ~\cite[Thm. 1.13]{Reid80} (see also
~\cite[Prop. 7.7]{KollarPairs}), the general fiber of
$h_d$ is a reduced, normal, local complete intersection stack having
at worst canonical singularities.  The one issue that remains is
connectedness, i.e. it is a priori possible that $h_d:C_d \rightarrow
\PP H^0(\PP(V), \OO_{\PP(V)}(d) )$ has a nontrivial Stein
factorization.  But observe that by Corollary~\ref{thm-e=1} (in fact
by the part reproved there), the restriction of $h_d$ to $Y$ is
surjective and has a trivial Stein factorization.  So $h_d|_Y$ yields
a section of the Stein factorization of $h_d$, which is irreducible
and finite over $\PP H^0(\PP(V), \OO_{\PP(V)}(d) )$.  It follows that
also the Stein factorization of $h_d$ is trivial, i.e. the fibers of
$h_d$ are connected.  So the general fiber of $h_d$ is an integral,
normal, local complete intersection stack of the expected dimension
having at worst canonical singularities.
\end{proof}

\section{Singularities of $\kbm{0,0}{\PP^n,e}$}
~\label{sec-RSBT} 

\medskip\noindent
In this section the Reid--Shepherd-Barron--Tai criterion is used
to prove that (with a very few exceptions) the coarse moduli spaces
$\kbm{0,0}{\PP^n,e}$ are terminal.  The same computations are used to
prove that if one carries out the deformation to the normal cone
construction as in Section~\ref{sec-proof} over the coarse moduli
space (instead of the stack, as in the last section), 
one obtains a family whose general fiber is the coarse moduli
space of $C_d$ and whose special fiber is a normal, $\QQ$-Gorenstein,
canonical variety.  The ``inversion of adjunction''
conjecture then implies that $C_d$ is itself canonical, and therefore
$\kbm{0,0}{X,e}$ is canonical for $X$ general.

\medskip\noindent
Let $\Gamma$ be a finite cyclic group of order $r$ and let $\zeta \in
\text{Hom}_{\text{group}}(\Gamma, \mathbb{G}_m)$ be a generator for
the character group of $\Gamma$.  Let $M$ be a finite dimensional
$\Gamma$-representation over $k$ (the field $K$ is still of
characteristic $0$, but the following definition makes sense so long as the
characteristic is prime to $\#\Gamma$).  There is a direct sum
decomposition,
\begin{equation}
M = \oplus_{i=0}^{r-1} L_{\zeta^i}^{\oplus a_i},
\end{equation}
where each $L_{\zeta^i}$ is the one-dimensional representation
corresponding to the character $\zeta^i$.  

\begin{defn} \label{defn-invt} 
The \emph{invariant of} $M$ \emph{with respect to} $\zeta$ (after
Reid--Shepherd-Barron--Tai) is
\begin{equation}
\alpha_\zeta(M) = \frac{1}{r} \sum_{i=0}^{r-1} ia_i.
\end{equation}
The \emph{invariant of} $M$ is $\alpha(M) = \min{\alpha_\zeta(M)}$ as
$\zeta$ varies over all generators of the character group.
\end{defn}

The importance of the invariant is the following theorem.

\medskip\noindent
\begin{thm} [Reid--Shepherd-Barron--Tai criterion, ~\cite{YPGS}]
    \label{thm-RSBT} 
Let $Y$ be a smooth $k$-variety, let $G$ be a finite subgroup of the
group of $k$-automorphisms of $Y$, and suppose that $G$ acts without
quasi-reflections.  Then the quotient variety $X=Y//G$ is terminal
(resp. canonical) iff for every cyclic subgroup $\Gamma \subset G$ and
every closed point $x\in Y^\Gamma$, the invariant of the Zariski
tangent space to $Y$ at $x$ satisfies
\begin{equation}
\alpha(T_x Y) > 1, \ \ \text{resp.} \ \ \alpha(T_x Y) \geq 1.
\end{equation}
\end{thm}

\medskip\noindent
\begin{cor} \label{cor-RSBT} 
Let $\mc{X}$ be a smooth, connected Deligne-Mumford stack over $K$.
Denote by $p:\mc{X} \rightarrow X$ the coarse moduli space, and
suppose that $p$ is an isomorphism over the complement of a closed
subset of codimension $\geq 2$.  Then $X$ is terminal
(resp. canonical) iff for every geometric point $x$ of $\mc{X}$, and
for every cyclic subgroup $\Gamma$ of the stabilizer group of $x$, the
invariant of the Zariski tangent space to $\mc{X}$ at $x$ satisfies
\begin{equation}
\alpha(T_x \mc{X}) > 1, \ \ \text{resp.} \ \ \alpha(T_x Y) \geq 1.
\end{equation}
\end{cor}

\medskip\noindent
\begin{proof}
This is just a rewording of Theorem~\ref{thm-RSBT} into the language
of Deligne-Mumford stacks.
\end{proof}

\medskip\noindent
\begin{cor} \label{cor-DMsmooth} 
Let $\mc{X}$ be a smooth, connected Deligne-Mumford stack over $K$.  Let
$f:\mc{Y} \rightarrow \mc{X}$ be a smooth, representable $1$-morphism
of Deligne-Mumford stacks.  Suppose that the morphism $p:\mc{X}
\rightarrow X$ is an isomorphism away from codimension $2$ and that
$X$ is terminal (resp. canonical).  Then the map to the coarse moduli
space of $\mc{Y}$, say $q:\mc{Y} \rightarrow Y$, is an isomorphism
away from codimension $2$ and $Y$ is terminal (resp. canonical).
\end{cor}

\medskip\noindent
\begin{proof}
Let $U\subset \mc{X}$ denote the maximal open substack over which $p$
is an isomorphism.  Then $\mc{X} - U$ has codimension at least $2$ in
$\mc{X}$.  Since $f$ is smooth, in particular it is flat.  Therefore
$f^{-1}(\mc{X} -U)$ has codimension at least $2$ in $\mc{Y}$.  And
$f^{-1}(U)$ is a scheme because $f$ is representable.  Therefore $q$
is an isomorphism when restricted to $f^{-1}(U)$, which shows that $q$
is an isomorphism away from codimension $2$.

\medskip\noindent
Next, apply Corollary~\ref{cor-RSBT}.  Let $y$ be a geometric point
of $\mc{Y}$ and let $x=f(y)$.  Because $f$ is representable, the
homomorphism from the stabilizer group of $y$ to the stabilizer group
of $x$ is injective.  So a cyclic subgroup $\Gamma$ of the stabilizer
group of $y$ is also a cyclic subgroup of the stabilizer group of $x$.
By Corollary~\ref{cor-RSBT}, the invariant of $T_x \mc{X}$ as a
$\Gamma$-representation is greater than $1$ (resp. at least $1$).
Since $f$ is smooth, the differential $df:T_y \mc{Y} \rightarrow T_x
\mc{X}$ is surjective.  Therefore the invariant of $T_y \mc{Y}$ is
greater than or equal to the invariant of $T_x \mc{X}$.  Applying
Corollary~\ref{cor-RSBT} one more time, $Y$ is
terminal (resp. canonical).
\end{proof}

\medskip\noindent
\begin{rmk} \label{rmk-DMsmooth} 
Unfortunately, for \emph{nice} representable $1$-morphisms $f:\mc{Y}
\rightarrow \mc{X}$ of smooth Deligne-Mumford stacks that are not
smooth, Corollary~\ref{cor-DMsmooth} often fails.  For instance, if
$\mc{Z} \subset \mc{X}$ is a Zariski closed substack that is smooth
and $f:\mc{Y} \rightarrow \mc{X}$ is the blowing up of $\mc{X}$ along
$\mc{Z}$, it can happen that $p:\mc{X} \rightarrow X$ is an
isomorphism away from codimension $2$, that $q:\mc{Y} \rightarrow Y$
is an isomorphism away from codimension $2$, that $X$ is terminal
(resp. canonical), but $Y$ is not terminal (resp. canonical).  For
instance, consider the action of the group of third roots of unity
$\mathbb{\mu}_3$ on affine $4$-space $\AAA^4$ by $\omega \cdot
(X_1,X_2,X_3,X_4) = (\omega X_1, \omega X_2, \omega X_3, \omega X_4)$.
Let $Z \subset \AAA^4$ be the variety associated to the invariant
ideal $\langle X_1, X_2, X_3 \rangle$.  Let $Y\rightarrow \AAA^4$
denote the blowing up along $Z$.  Then $f:[Y/\mathbb{\mu}_3]
\rightarrow [X/\mathbb{\mu}_3]$ is a $1$-morphism of smooth
Deligne-Mumford stacks satisfying the hypotheses above and
$X//\mathbb{\mu}_3$ is terminal.  But $Y//\mathbb{\mu}_3$ is not even
canonical.
\end{rmk}


\medskip\noindent
Let $\Gamma$ be a finite cyclic group of order $r$ and let $\Delta
\subset \Gamma$ be a subgroup of index $s$.  Let $\gamma:\Gamma
\rightarrow \mathbb{G}_m$ be a generator for the character group of
$\Gamma$.  The restriction of $\gamma$ to $\Delta$ is a generator for
the character group of $\Delta$.  The following lemma is a rewording
of the argument on pp. $33$--$34$ of ~\cite{HM}.

\medskip\noindent
\begin{lem}~\cite[pp. $33$--$34$]{HM} \label{lem-induced} 
Let $V$ be a finite-dimensional representation of $\Delta$ and let $V
\otimes_{K[\Delta]} K[\Gamma]$ be the induced $\Gamma$-representation.
The relation between the invariant of $V\otimes_{K[\Delta]} K[\Gamma]$
as a $\Gamma$-representation and the invariant of $V$ as a
$\Delta$-representation is,
\begin{equation}
\alpha_{\Gamma,\gamma}\lt( V\otimes_{K[\Delta]} K[\Gamma] \rt) =
\alpha_{\Delta,\gamma|_{\Delta}}(V) + \frac{s-1}{2}\text{dim}_K(V).
\end{equation}
\end{lem}

\medskip\noindent
\begin{proof}
Each side of the equation is additive in $V$, therefore it suffices
to consider the case that $V$ is an irreducible representation, i.e. a
character $V=L_{\gamma|_{\Delta}^l}$ for some integer $l=0,\dots
\frac{r}{s}-1$.  Let $\phi$ be a generator for $\Gamma$, so that
$\phi^s$ is a generator for $\Delta$.  Let $\overline{\epsilon}$ be a
nonzero element of $V$.  For each integer $j=0,\dots,s-1$, denote
$m=-l-\frac{j\cdot r}{s}$ and define the element $\epsilon_j \in V
\otimes_{K[\Delta]} K[\Gamma]$ to be,
\begin{equation}
\epsilon_j = \sum_{i=0}^{s-1} \gamma^{m}(\phi^i)
\overline{\epsilon}\otimes \phi^i.
\end{equation}
It is trivial to compute that $\epsilon_j \cdot \phi =
\epsilon_j \cdot \gamma^{-m}(\phi)$.  So $\epsilon_j$ spans an
irreducible subrepresentation of $V \otimes{K[\Delta]} K[\Gamma]$
that is isomorphic to $L_{\gamma^{-m}}$.  

\medskip\noindent
This gives $s$ different irreducible subrepresentations of
$V\otimes_{K[\Delta]} K[\Gamma]$, which is also the dimension as a
$K$-vector space.  So there is an irreducible decomposition,
\begin{equation}
V\otimes_{K[\Delta]} K[\Gamma] \cong \bigoplus_{j=0}^{s-1}
L_{\gamma^{l + j\cdot\frac{r}{s}}}.
\end{equation}
It follows that the invariant of $V\otimes_{K[\Delta]} K[\Gamma]$ as a
$\Gamma$ representation is
\begin{equation}
\begin{array}{r}
\frac{1}{r} \lt[l + (l+\frac{r}{s}) + \dots + (l +
(s-1)\frac{r}{s})\rt] =
l/\lt({\frac{r}{s}}\rt) + \frac{s-1}{2} = \\
 \alpha_{\Delta,\gamma|_{\Delta}}(V)
+ \frac{s-1}{2} \text{dim}_K(V).
\end{array}
\end{equation} 
\end{proof}

\medskip\noindent
Recall that $\Kbmo{0,0}{\PP^n,e} \subset \Kbm{0,0}{\PP^n,e}$ is the
open substack parametrizing stable maps with irreducible domain, and
$\kbmo{0,0}{\PP^n,e}$ is the coarse moduli space of
$\Kbmo{0,0}{\PP^n,e}$.

\begin{prop} \label{prop-RSBT1} 
Let $x$ be a geometric point of $\Kbmo{0,0}{\PP^n,e}$ and let $\Gamma$
be a subgroup of the stabilizer group of $x$.  Denote $r = \#\Gamma$.
The invariant of the Zariski tangent space to $\Kbmo{0,0}{\PP^n,e}$ at
$x$ equals
\begin{equation}
\alpha(T_x \Kbmo{0,0}{X,e}) = \frac{e(n+1)}{2} \lt( 1 - \frac{1}{r}
\rt) - 1.
\end{equation}
Except in the cases $(e,n) = (2,1)$ and $(e,n) = (2,2)$, the map
$p:\Kbmo{0,0}{\PP^n,e} \rightarrow \kbmo{0,0}{\PP^n,e}$ is an
isomorphism away from codimension $2$.  Except for the cases $(e,n)=(2,1),
(2,2), (3,1)$, and $(2,3)$, $\kbmo{0,0}{\PP^n,e}$ is terminal. 
In the the cases
$(e,n) = (3,1), (2,3),$ $\kbmo{0,0}{\PP^n,e}$ is canonical.
\end{prop} 

\medskip\noindent
\begin{proof}
By the same $\text{GL}_{n+1}$-invariance argument as in the proof of
Theorem~\ref{thm-main}, it suffices to prove the result when $x$ is a
geometric point of $Y \cap \Kbmo{0,0}{\PP^n,e}$.  At such a point 
the Zariski tangent space decomposes as a direct sum of the Zariski
tangent space to $Y$ and the normal bundle to $Y$.  The Zariski
tangent space to $Y$ further decomposes as the direct sum of the
Zariski tangent space to $\mathbb{G}$ and the vertical tangent bundle
of $\text{pr}_{\mathbb{G}}:Y \rightarrow \mathbb{G}$.  And by
Lemma~\ref{lem-phi1}, the normal bundle to $Y$ is a direct sum of
$n-1$ copies of $\mc{R}^\vee$.  What is needed is to compute the invariants
(with respect to some $\zeta$) of the vertical tangent bundle of
$\text{pr}_{\mathbb{G}}$ and of $\mc{R}^\vee$.

\medskip\noindent
The vertical tangent bundle of $\text{pr}_{\mathbb{G}}$ is the
same as the tangent bundle of $\Kbmo{0,0}{\PP^1,e}$, so suppose now
that $n=1$.  Let the geometric point $x$ parametrize a stable map $f:C
\rightarrow \PP^1$.  Choose a generator for $\Gamma$, which will be an
automorphism $\phi:C \rightarrow C$ such that $f\circ \phi =f$ and
such that $\phi^s = \text{Id}$ iff $r$ divides $s$.  It is easy to
show that, up to a choice of homogeneous coordinates, $\phi:C
\rightarrow C$ is just the isomorphism $[X_0:X_1] \mapsto [X_0:\xi
X_1]$ for some primitive $r^{\text{th}}$ root of unity.

\medskip\noindent
Denote by $g:C \rightarrow C_0$ the quotient of $C$ by $\phi$ and let
$h:C_0 \rightarrow \PP^1 $ be the unique morphism such that $f=h\circ
g$.  The Zariski tangent space to $\Kbmo{0,0}{\PP^1,e}$ is just the
vector space of global sections of the torsion sheaf
$f^*T_{\PP^1}/T_C$.  And this fits into an exact sequence,
\begin{equation}
\begin{CD} 
0 @>>> g^* T_{C_0}/T_C @>>> g^* h^* T_{\PP^1}/T_C @>>> g^*\lt(
h^*T_{\PP^1}/T_{C_0} \rt) @>>> 0.
\end{CD}
\end{equation}
Now, as a representation of $\Gamma$, $g^* \lt( h^* T_{\PP^1}/T_{C_0}
\rt)$ is isomorphic to the tensor product $\lt( h^* T_{\PP^1}/T_{C_0}
\rt) \otimes_K K[\Gamma]$, where the first factor is a trivial
representation.  In particular, the invariant with respect to any
generator $\zeta$ is just,
\begin{equation}
\alpha_\zeta(g^* ( h^* T_{\PP^1}/T_{C_0} ) ) = 2(\frac{e}{r} -1) \cdot
\lt( \frac{0}{r} + \frac{1}{r} + \dots + \frac{r-1}{r} \rt) =
(e-r)\lt( 1 - \frac{1}{r} \rt).
\end{equation}

\medskip\noindent
By direct computation, as a representation of $\Gamma$,
$g^*T_{C_0}/T_C$ is isomorphic to
\begin{equation}
g^* T_{C_0}/T_C \cong L_{\xi^0}^{\oplus 2} \oplus L_{\xi^1}^{\oplus 1}
\oplus L_{\xi^{r-1}}^{\oplus 1} \oplus \bigoplus_{i=1}^{r-2}
L_{\xi^i}^{\oplus 2}.
\end{equation}
It follows that the invariant with respect to any generator $\zeta$ is
just,
\begin{equation}
\alpha_\zeta( g^* T_{C_0}/T_C) = 2 \lt(\frac{0}{r} + \frac{1}{r} +
\dots + \frac{r-1}{r} \rt) - \lt( \frac{i}{r} + \frac{r-i}{r} \rt) =
r\lt( 1 - \frac{1}{r} \rt) - 1.
\end{equation}
Here $i$ is the unique integer such that $\{ \xi^1, \xi^{r-1} \} = \{
\zeta^i, \zeta^{r-i} \}$.  Summing up, with respect to
any generator $\zeta$,
\begin{equation}
\alpha_\zeta( T_x \Kbmo{0,0}{\PP^1,e} ) = e\lt( 1 - \frac{1}{r} \rt) -
1.
\end{equation}

\medskip\noindent
Next consider the invariant of $\mc{R}|_x$.  For clarity, 
denote this vector
space by $\mc{R}_f$.  Each of $g:C \rightarrow C_0$ and $h:C_0
\rightarrow \PP^1$ is also a stable map of a genus $0$ curve to
$\PP^1$.  So each of these also has a canonically associated vector
space $\mc{R}_g$, respectively $\mc{R}_h$.  As $\Gamma$
representations, $\mc{R}_g$ is just $K[\Gamma]/K{1}$ (by direct
computation), and $\mc{R}_h$ is a trivial representation of dimension
$\frac{e}{r} - 1$.  It is easy to see that the relationship between
these spaces is that $\mc{R}_f$ is isomorphic as a
$\Gamma$-representation to $\mc{R}_h \otimes_K \mc{R}_g \oplus
\mc{R}_g \oplus \mc{R}_h$.  Therefore the invariant of $\mc{R}_f$ is
\begin{equation}
\alpha_\zeta( \mc{R}_f ) = \frac{e}{2}\lt( 1 - \frac{1}{r} \rt).
\end{equation}
This is also the invariant of $\mc{R}_f^\vee$.

\medskip\noindent
As mentioned above, the normal bundle of $Y$ at $x$ is isomorphic as a
$\Gamma$-representation to a direct sum of $n-1$ copies of
$\mc{R}_f^\vee$.  And the vertical tangent space to
$\text{pr}_{\mc{G}}$ is the same as the tangent space to
$\Kbm{0,0}{\PP^1,e}$.  Therefore,
\begin{equation}
\alpha_{\zeta}(T_x \Kbmo{0,0}{\PP^n,e}) = \frac{(n-1)e}{2} \lt( 1 -
\frac{1}{r} \rt) + e\lt( 1 - \frac{1}{r} \rt) - 1 = \frac{(n+1)e}{2}
\lt( 1 - \frac{1}{r} \rt) - 1.
\end{equation}

\medskip\noindent
For $(e,n) \neq (2,1), (2,2)$, the invariant is at least $1$, 
proving that the stabilizer group of $x$ acts without quasi-reflections
and the coarse moduli space has canonical singularities.  Moreover,
except in the extra cases $(e,n) = (3,1), (2,3)$, the invariant is
actually larger than $1$, proving that the coarse moduli space has
terminal singularities.
\end{proof}

\medskip\noindent
\begin{rmk} \label{rmk-RSBT} 
In case $e=2, \ n=1$, every geometric point of $\Kbmo{0,0}{\PP^1,2}$
has nontrivial stabilizer.  In fact the coarse moduli space
$\kbmo{0,0}{\PP^1,2}$ is isomorphic to the complement of a smooth
plane conic in $\PP^2$ (via the \emph{branch morphism},
c.f. ~\cite{FaP}), and $p:\Kbmo{0,0}{\PP^1,2} \rightarrow
\kbmo{0,0}{\PP^1,2}$ is a $\ZZ/2\ZZ$-gerbe.  In case $e=2, \ n=2$, the
coarse moduli space $\kbmo{0,0}{\PP^2,2}$ is smooth and is isomorphic
to an open subset of the blowing up of $\PP^5$ along a Veronese
surface (the open subset is the complement of the proper transform of
the discriminant hypersurface).  In this case the morphism
$p:\Kbmo{0,0}{\PP^2,2} \rightarrow \kbmo{0,0}{\PP^2,2}$ is an
isomorphism on the complement of the exceptional divisor, and over the
exceptional divisor it is a $\ZZ/2\ZZ$-gerbe.
\end{rmk}

\medskip\noindent
\begin{lem} \label{lem-Gor} 
Let $(e,n)\neq (2,1), (2,2),$ be a pair of positive integers. 
If at least one of $e$ and $n$ is odd, then $\kbmo{0,0}{\PP^n,e}$ is
Gorenstein.  If both $e$ and
$n$ are even, then $\kbmo{0,0}{\PP^n,e}$ is not Gorenstein.
\end{lem}

\medskip\noindent
\begin{proof}
Since $(e,n) \neq (2,1), (2,2)$, $p:\Kbmo{0,0}{\PP^n,e}
\rightarrow \kbmo{0,0}{\PP^n,e}$ is an isomorphism away from
codimension $2$.  It follows from ~\cite[Prop. 5.75]{KM} that the
dualizing sheaf of $\kbmo{0,0}{\PP^n,e}$ is the pushforward by $p_*$
of the dualizing sheaf of $\Kbmo{0,0}{\PP^n,e}$.  Given a geometric
point $x$ of $\Kbmo{0,0}{\PP^n,e}$, the dualizing sheaf of
$\kbmo{0,0}{\PP^n,e}$ is invertible at $p(x)$ iff there exists a
section of the dualizing sheaf near $p(x)$ whose pullback to
$\Kbmo{0,0}{\PP^n,e}$ is non-zero at $x$.  Such a section corresponds
to a nonzero element in the one-dimensional vector space
$\text{det}(T_x \Kbmo{0,0}{\PP^n,e})^\vee$ that is invariant under
the action of the stabilizer group of $x$.  Therefore
$\kbmo{0,0}{\PP^n,e}$ is Gorenstein iff for every geometric point $x$
of $\Kbmo{0,0}{\PP^n,e}$ and for every cyclic subgroup $\Gamma$ of the
stabilizer group of $x$, the induced character $\text{det}(T_x
\Kbmo{0,0}{\PP^n,e})$ is trivial.

\medskip\noindent
As in the proof of Proposition~\ref{prop-RSBT1}, it suffices to
compute the character for geometric points $x$ of $Y\cap
\Kbmo{0,0}{\PP^n,e}$.  In the proof of Proposition~\ref{prop-RSBT1} 
the characters of all relevant $\Gamma$-representations were computed.  
The character of
$g^*\lt( h^*T_{\PP^1}/T_{C_0} \rt)$ is just
$\text{det}(K[\Gamma])^{\otimes 2(\frac{e}{r} -1)}$ where
$2(\frac{e}{r} -1)$ is the dimension of $h^*T_{\PP^1}/T_{C_0}$.
Similarly, the character of $T_x \Kbmo{\PP^1,e}$ is
$\text{det}(K[\Gamma])^{\otimes 2}$ (the missing $L_{\xi^1}$ and
$L_{\xi^{r-1}}$ factors tensor to give a trivial character).  Finally,
the character of $\mc{R}_g$ is $\text{det}(K[\Gamma])$ and the
character of $\mc{R}_f \cong \lt( \mc{R}_h \otimes_K \mc{R}_g \rt)
\oplus \lt( \mc{R}_h \rt) \oplus \lt( \mc{R}_g \rt)$ is
$\text{det}(K[\Gamma])^{\otimes \frac{e}{r} }$.  Altogether, the
character of $T_x \Kbmo{0,0}{\PP^n,e}$ is
$\text{det}(K[\Gamma])^{\otimes (n+1)\frac{e}{r}}$.

\medskip\noindent
If $r$ is odd, then the character $\text{det}(K[\Gamma])$ is trivial
so that the character of $T_x \Kbmo{0,0}{\PP^n,e}$ is trivial.  But if
$r$ is even, the character $\text{det}(K[\Gamma])$ equals
$L_{\zeta^{\frac{r}{2}}}$ for any generator $\zeta$ of the character
group of $\Gamma$.  This is a nontrivial character whose square is
trivial.  For $r$ even, the character of $T_x \Kbmo{0,0}{\PP^n,e}$ is
nontrivial iff $\frac{e}{r}$ is odd and $n+1$ is odd.  Therefore if
$e$ is odd or $n$ is odd, then the character of $T_X
\Kbmo{0,0}{\PP^n,e}$ is trivial for every geometric point,
i.e. $\kbmo{0,0}{\PP^n,e}$ is Gorenstein.

\medskip\noindent
On the other hand, suppose that $n$ and $e$ are both even.  Then for
any line $L \subset \PP^n$ and any reduced degree $2$ divisor on $L$,
the cyclic cover $f:C \rightarrow L$ of degree $e$ branched over that
divisor gives a stable map of degree $e$ whose stabilizer group is
cyclic of order $r=e$.  Therefore the character of $T_x
\Kbmo{0,0}{\PP^n,e}$ is nontrivial, i.e. $\kbmo{0,0}{\PP^n,e}$ is not
Gorenstein.
\end{proof}

\medskip\noindent
\begin{prop} \label{prop-RSBT2} 
Let $(e,n)\neq (2,1),(2,2),$ be a pair of positive integers. 
If $e \geq 3$ and $n=1$, the coarse moduli space
$\kbm{0,0}{\PP^1,e}$ is canonical.  If $(e,n)=(2,3)$, the coarse
moduli space $\kbm{0,0}{\PP^3,2}$ is canonical.  In all other cases,
the coarse moduli space $\kbm{0,0}{\PP^n,e}$ is terminal.
\end{prop}

\medskip\noindent
\begin{proof}
The proof follows closely the argument on pp. $33$--$34$ of
~\cite{HM}.  As in the proof of Proposition~\ref{prop-RSBT1}, to prove
that $\kbm{0,0}{\PP^n,e}$ is canonical (resp. terminal), it suffices
to check that for every geometric point $x$ of $Y$ and for every
cyclic subgroup $\Gamma$ of the stabilizer group of $x$, the invariant
of $T_x \Kbm{0,0}{\PP^n,e}$ is bigger than $1$ (resp. at least $1$).
Choose some line $\PP^1 \subset \PP^n$; the
geometric point $x$ will belong to 
the closed substack $\Kbm{0,0}{\PP^1,e}
\subset \Kbm{0,0}{\PP^n,e}$.  The proof that the invariant
of $T_x \Kbm{0,0}{\PP^n,e}$ is bigger than $1$ (resp. at least $1$)
proceeds by
induction on the number $\delta$ of nodes of $C$.  
The base case is when $\delta=0$, i.e. $x$ is in
$\kbmo{0,0}{\PP^n,e}$, and follows from Proposition~\ref{prop-RSBT1}.
Therefore suppose $\delta > 0$,
and, by way of induction, suppose the result has been
proved for all points with fewer than $\delta$ nodes.

\medskip\noindent
Let $\phi$ be a generator for $\Gamma$ and let $\{q,\phi q, \phi^2 q,
\dots, \phi^{s-1} q\}$ be an orbit of $\Gamma$ on $C$ such that each
$\phi^i q$ is a node.  Of course, $s$ divides the order of $\Gamma$,
which is denoted by $r$.  
The language of ~\cite{BM} for stable $A$-graphs is used.  Denote by
$\tau$ the stable $A$-graph of $f:C \rightarrow \PP^1$,
i.e. $\tau$ is the dual
graph of $C$ labelled by the degree of $f$, and denote by
$\Kbm{}{\PP^1,\tau}$ the corresponding
Behrend-Manin moduli stacks (essentially this is the closed substack
of $\Kbm{0,0}{\PP^1,e}$ parametrizing stable maps obtained as
specializations of deformations of $f$ that do not smooth the nodes of
$C$).  
Let $E_0,E_1,\dots, E_{s-1}$
denote the edges of $\tau$ corresponding to the nodes $q, \phi q, \dots,
\phi^{s-1} q$.  Let $\psi: \tau \rightarrow \sigma$ be the maximal
contraction of $\tau$ not contracting any of the edges $E_0,
\dots, E_{s-1}$, i.e. $\sigma$ is the same as the dual graph of a
curve obtained by smoothing all the nodes of $C$ except $q,\dots,
\phi^{s-1}q$.  Then $f:C \rightarrow \PP^1$ gives a geometric point of
the Behrend-Manin moduli stack $\Kbm{}{\PP^1,\sigma}$, i.e. the moduli
space of stable maps to $\PP^1$ whose dual graph has a contraction to
$\sigma$.

\medskip\noindent
There is a canonical $1$-morphism $\Kbm{}{\PP^1,\sigma} \rightarrow
\Kbm{0,0}{\PP^1,e}$ that is unramified and whose normal sheaf is
locally free.  Therefore the tangent bundle of $\Kbm{}{\PP^1,\sigma}$
at $[f:C \rightarrow X]$ is a vector subspace of the tangent bundle of
$\Kbm{0,0}{\PP^1,e}$ at $[f:C \rightarrow X]$.  Moreover the cokernel,
i.e. the normal bundle, is precisely,
\begin{equation}
N_{[f]} = \bigoplus_{i=0}^{s-1} T'_{\phi^i q} \otimes_k T''_{\phi^i q},
\end{equation}
where $T'_{\phi^i q}$ and $T''_{\phi^i q}$ are the tangent spaces of
the two branches of $C$ at $\phi^i q$ (there isn't any canonical
ordering of the two branches; the notation $T'$ and $T''$ is just for
convenience).  

\medskip\noindent
Now suppose there exists a nonzero $\Gamma$-invariant
section $\epsilon \in N_{[f]}$.  Then there exists a
$\Gamma$-invariant
section
$\widetilde{\epsilon}$ of $T_[f] \Kbm{0,0}{\PP^1,e}$ 
mapping to $\epsilon$.  Over an \'etale
neighborhood of the image of $[f]$ in $\kbm{0,0}{\PP^1,e}$, the stack
$\Kbm{0,0}{\PP^1,e}$ is a finite group quotient $[M/G]$, where $G$ is
the stabilizer group of $f$ and $M$ is a smooth scheme.  In particular
the invariant locus $M^\Gamma \subset M$ is a smooth closed subscheme 
whose Zariski tangent space at $[f]$ is the
$\Gamma$-invariant subspace of $T_{[f]} \Kbm{0,0}{\PP^1,e}$.  In
particular there exists a smooth, connected curve $B\subset M^\Gamma$
such that $B$ contains the point $[f]$ and the tangent space to $B$ at
$[f]$ equals $\text{span}(\widetilde{\epsilon})$.  Since
$\widetilde{\epsilon}$ has nonzero image in $N_{[f]}$, the curve $B$
is not contained in the image of $\Kbm{}{\PP^1,\sigma}$.  So a
general point of $B$ parametrizes a stable map with fewer nodes than
$f:C\rightarrow \PP^1$.  On the other hand, the invariant of the
Zariski tangent space is constant in connected families.  So by the
induction hypothesis, the invariant of $f$ equals the invariant of a
general point of $B$ which is greater than $1$ (resp. at least $1$).

\medskip\noindent
By the last paragraph, the proof reduces to the case when 
the $\Gamma$-invariant subspace of $\bigoplus_{i=0}^{s-1}
T'_{\phi^i q} \otimes T''_{\phi^i q}$ is trivial for every node $q$ of
$C$.  Let $\Delta \subset
\Gamma$ denote the subgroup generated by $\phi^s$.  There is an action of
$\Delta$ on $T'_q \otimes T''_q$, and the $\Gamma$-representation
$N_{[f]}$ is simply the induced representation $\lt( T'_q \otimes
T''_q \rt) \otimes_{K[\Delta]} K[\Gamma]$.  By
Lemma~\ref{lem-induced}, the invariant of $N_{[f]}$ as
a $\Gamma$-representation is simply,
\begin{equation}
\alpha_\gamma(N_{[f]}) = l\cdot \frac{s}{r} + \frac{s-1}{2},
\end{equation}
where the character $T'_q\otimes T''_q$ of $\Delta$ is
$\gamma|_{\Delta}^l$ for $l=0,\dots, \frac{r}{s}-1$.  


\medskip\noindent
If $s \geq 3$, then already the invariant of $N_{[f]}$ is greater than
$1$.  Thus assume that $s=1$ or $s=2$.  If $s=2$, then the
invariant of $N_{[f]}$ is $\frac{1}{2} + l \cdot \frac{2}{r}$.  If
$s=1$, then the invariant is $l \cdot \frac{1}{r}$.  The only
possibilities that don't give an invariant larger than $1$ are,
\begin{enumerate}
\item[(i)] 
every node of $C$ is fixed by $\phi$, or
\item[(ii)] 
there is precisely one pair of nodes $q, \phi q$ not fixed by
  $\phi$.
\end{enumerate}
These possibilities are considered in turn.

\medskip\noindent
\textbf{(i)}:
Suppose first that every node of $C$ is fixed by $\phi$.  Then every
irreducible component of $C$ is stabilized by $\phi$ (as a set, not
pointwise).  Since
$\phi$ is nontrivial, there exists an irreducible component $C_i$ of $C$
on which the action of $\phi$ is nontrivial.
Let $C_1$ be an irreducible
component so that the restriction $\phi|_{C_1}$ has maximal order
$r_1$ (i.e. $\phi|_{C_1}^m = \text{Id}$ iff $r_1$ divides $m$).  The
irreducible component $C_1$ contains at least one node of $C$ and
contains no more than two nodes of $C$ since the only
automorphism of $\PP^1$ fixing three points is the identity.  In
particular, $C_1$ is not contracted by $f$.

\medskip\noindent
Let $\tau$ be the stable $A$-graph of $C$ and let $\psi:\tau
\rightarrow \sigma$ be the maximal contraction not contracting
the edges corresponding to nodes on $C_1$.  There is a morphism
$\Kbm{}{\PP^1,\sigma} \rightarrow \Kbm{0,0}{\PP^1,e}$ and the normal
bundle, as mentioned above, is the direct sum over nodes $q$ on $C_1$
of $T'_q \otimes T''_q$.  Let $\xi: \sigma \hookrightarrow \tau'$ be
the combinatorial morphism that is the inclusion of the maximal
sub-$A$-graph of $\tau$ whose only vertex is $v_1$, the vertex of the
irreducible component of $C_1$.  More simply, $\tau'$ is the graph
with the single vertex $v_1$ corresponding to $C_1$ and with one tail
for each node of $C$ contained in $C_1$ (i.e. either one or two
tails depending on whether $C_1$ contains one or two nodes of $C$).
Let $f_1:(C_1,q) \rightarrow \PP^1$ or $f_1:(C_1,q,q') \rightarrow
\PP^1$ be the marked stable map that is the restriction of $f$ to the
irreducible component $C_1$ marked by the nodes of $C$ contained on
$C_1$.

\medskip\noindent
There is a commutative diagram of $1$-morphisms:
\begin{equation}
\begin{CD}
\Kbm{}{\sigma, \PP^1} @>>> \Kbm{}{\sigma,\PP^n} \\
@V \Kbm{}{\xi, \PP^1} VV @VV \Kbm{}{\xi, \PP^n} V \\
\Kbm{}{\tau', \PP^1} @>>> \Kbm{}{\tau', \PP^n}
\end{CD}
\end{equation}
The horizontal arrows are closed immersions and the vertical arrows
are smooth.  So the invariant of the tangent space $T_{[f]}
\Kbm{}{\sigma,\PP^n}$ is greater 
than or equal to the invariant of the tangent space $T_{[f_1]}
\Kbm{}{\tau',\PP^n}$.  Let $e_1$ be the degree of $f_1:C_1 \rightarrow
\PP^1$.  As a $\Gamma$-representation, $T_{[f_1]} \Kbm{}{\tau', \PP^n}$
is the direct sum of $T_{[f_1]} \Kbmo{0,0}{\PP^n,e_1}$ with the tangent
space $T_q C_1$ (or $T_q C_1 \oplus T_{q'} C_1$ if $C_1$ contains two
nodes).  By Proposition~\ref{prop-RSBT1}, the invariant of $T_{[f_1]}
\Kbmo{0,0}{\PP^n,e_1}$ is $\frac{e_1(n+1)}{2}\lt( 1- \frac{1}{r_1}
\rt) - 1$.  Except for the four cases $(e_1,n) = (2,1), (2,2), (2,3),
(3,2)$, this invariant is already greater than $1$, hence the invariant
of $T_[f] \Kbm{0,0}{\PP^n,e}$ is also greater than $1$.  
To finish the proof in case (i), each case $(e_1,n) = (2,1), (2,2),
(2,3), (3,2)$ is considered.

\medskip\noindent
\textbf{(i); $(e_1,n) = (2,3), (3,2)$}:
If $(e_1,n) = (2,3)$ or $(3,2)$ the invariant
$\frac{e_1(n+1)}{2}\lt( 1- \frac{1}{r_1} \rt) - 1$ equals $1$.  And
then the invariant of $T_q C$ is either $\frac{1}{2}$ for $(2,3)$ or
$\frac{1}{3}$ or $\frac{2}{3}$ for $(3,2)$.  Therefore the invariant
of $\Kbm{}{\tau', \PP^n}$ is greater than $1$.  So the invariant of
$T_{[f]} \Kbm{0,0}{\PP^n,e}$ is also greater than $1$.

\medskip\noindent
\textbf{(i); $(e_1,n) = (2,2)$}:
If $(e_1,n) = (2,2)$, the invariant of $T_{[f_1]} \Kbmo{0,0}{\PP^2,2}$
is $\frac{1}{2}$.  The invariant of $T_q C_1$ is also $\frac{1}{2}$.
So if there are two nodes on $C_1$, the invariant is already greater
than $1$.  If there is only one node, so far the invariant only
equals $1$.  But also the invariant of $T'_q \otimes T''_q$,
is positive.  So the invariant of $T_{[f]} \Kbm{0,0}{\PP^n,e}$
is greater than $1$.

\medskip\noindent
\textbf{(i); $(e_1,n)=(2,1)$}:
Finally, suppose $(e_1,n) = (2,1)$.  The invariant of $T_{[f_1]}
\Kbmo{0,0}{\PP^1,2}$ is zero.  The invariant of $T_q C_1$ is
$\frac{1}{2}$.  If there are two nodes on $C_1$, then the invariant is
$1$ and then the contributions of $T'_q\otimes T''_q$ and $T'_{q'}
\otimes T''_{q'}$ will make the total invariant of $T_{[f]}
\Kbm{0,0}{\PP^1,e}$ greater than $1$.  Therefore assume there is only one
node.  Then the invariant of $T_{[f_1]} \Kbm{}{\PP^1,\sigma}$ 
equals $\frac{1}{2}$.  Consider the $\Gamma$-representation $T'_q
\otimes T''_q$.  Let $C_2$ denote the irreducible component of $C$
that intersects $C_1$ at $q$.  Because the
order $r_1$ is the maximal among all orders of $\phi|_{C_i}$, either
$\phi$ acts trivially on $C_2$ or the order of $\phi|_{C_2}$ is $2$.
In the second case, both $T'_q$ and $T''_q$ give characters of
$\Gamma$ that are $\gamma^{\frac{r}{2}}$, the unique character of
order $2$.  So the tensor product is the trivial character.  This
violates the assumption that for every node there are no non-zero
$\Gamma$-invariant sections of $\bigoplus_{i=0}^{s-1} T'_{\phi^i q}
\otimes T''_{\phi^i q}$.  Therefore $\phi$ acts trivially on $C_2$ and
the invariant of $T'_q \otimes T''_q$ is $\frac{1}{2}$.  So the
invariant of $T_{[f]} \Kbm{0,0}{\PP^1,e}$ is at least $\frac{1}{2} +
\frac{1}{2} = 1$.  Unfortunately, this is the best one can do -- it is
quite easy to write down a degree $e \geq 3$ cover of $\PP^1$ with reducible
domain where the invariant equals $1$.  So for $n=1$ and $e\geq
3$, the conclusion is that the invariant is $\geq 1$,
i.e. $\Kbm{0,0}{\PP^1,e}$ has canonical singularities (of course we
still have to dispense with the case that there are two nodes $q,\phi
q$ interchanged by $\phi$!).

\medskip\noindent
\textbf{(ii)}:
This finishes the analysis when $\phi$ fixes all nodes.  Next
suppose that there is exactly one pair of nodes $\{ q, \phi q\}$ which
are not fixed by $\phi$.  The node $q$ disconnects $C$ into a union of
two connected subcurves, denoted $D_q$ and $C_2$.  Let $C_2$ denote 
the subcurve
containing $\phi q$.  The node $\phi q$ disconnects $C_2$ into a
union of two connected subcurves, denoted 
$D_{\phi q}$ and $C_1$.  Let $C_1$ denote 
the subcurve containing $q$.  So $C_1$ is the maximal connected
subcurve of $C$ containing $q$ and $\phi q$ on which both $q$ and
$\phi q$ are nonsingular points.  Observe that $\phi(D_q) = D_{\phi
q}$, $\phi(D_{\phi q}) = D_q$ and both $D_q$ and $D_{\phi q}$ are
smooth.  Let $C_q \subset C_1$ be the irreducible component 
containing $q$ and let $C_{\phi q} \subset C_1$ be the irreducible
component containing $\phi q$.  Observe that $\phi(C_q) = C_{\phi
q}$ and $\phi(C_{\phi q}) = C_q$.  There are two possibilities
depending on whether $C_q$ (and thus $C_{\phi q}$) is contracted by
$f$ or not.

\medskip\noindent
\textbf{(ii); $C_q$ is contracted}:
Suppose that $C_q$ is contracted by $f$ and
suppose that $C_q \neq C_{\phi q}$. Then $C_q$ contains at least three
nodes, $q$ and two other nodes.  By assumption, each of the two other
nodes is fixed by $\phi$.  Also $\phi(C_q) = \phi(C_{\phi
q})$.  Since $C_q$ is not equal to $C_{\phi q}$, then $C_q \cap
C_{\phi q}$ is at most one node.  But then the second of the other
nodes cannot be fixed by $\phi$.  This is a contradiction.  Therefore
if $C_q$ is contracted by $f$, then $C_q = C_{\phi q}$
and $C_q$ contains at least one other node $q'$ of $C_1$ that is one
of the two fixed points of $\phi|_{C_q}$.  Also, since $\phi|_{C_q}$
contains the orbit $\{q, \phi q\}$ of order $2$, 
$\phi|_{C_q}$ has order $2$.

\medskip\noindent
The node $q'$ disconnects $C_1$ into $C_q$ and a connected subcurve
$C_0$.  Suppose that $\phi|_{C_0}$ is the identity.  Then the
$\Gamma$-representation $T'_r \otimes T''_r$ has invariant
$\frac{1}{2}$.  Combined with the invariant $\frac{1}{2} + \frac{l
\cdot r}{2}$ coming from the nodes $\{ q, \phi q\}$, the total
invariant is greater than $1$ so that the invariant of $T_{[f]}
\Kbm{0,0}{\PP^n,e}$ is greater than $1$.  Next suppose that
$\phi|_{C_0}$ is not the identity.  Then by the same analysis as in (i),
the 
invariant coming from $C_0$ is at least $\frac{1}{2}$ (except when
$(e_1,n) = (2,1)$, in which case the invariant is at least $1$).  
Combined with the
invariant $\frac{1}{2} + \frac{l \cdot r} {2}$ coming from the nodes
$\{ q, \phi q\}$, the total invariant is greater than $1$ so that the
invariant of $T_{[f]} \Kbm{0,0}{\PP^n,e}$ is greater than $1$.

\medskip\noindent
\textbf{(ii); $C_q$ is not contracted}:
The final case is when $f$ does not contract $C_q$.  By the
same analysis as above, $D_q$ and $D_{\phi q}$ are
irreducible and are not contracted by $f$.  Let $\tau$ be the stable
$A$-graph of $f$ and let $\psi:\tau \rightarrow \sigma$ be the maximal
contraction not contracting the edges corresponding to the
nodes $q$ and $\phi q$.  Then $\sigma$ has three vertices: $v_0$
corresponding to the connected subcurve $D_q$, $v_1$ corresponding to
the connected subcurve $C_1$, and $v_2$ corresponding to the connected
subcurve $D_{\phi q}$.  The stable map $f:C \rightarrow \PP^1$
determines a point of the Behrend-Manin stack, $\Kbm{}{\PP^n,
\sigma}$.  The Zariski tangent space of $\Kbm{}{\PP^n, \sigma}$ at
$[f]$ is a $\Gamma$-sub-representation of the Zariski tangent space of
$\Kbm{0,0}{\PP^n,e}$ and the cokernel is $(T'_q \otimes T''_q) \oplus
(T'_{\phi q} \otimes T''_{\phi q})$.

\medskip\noindent
Let $\xi:\sigma \hookrightarrow \tau'$ be the maximal
\emph{disconnected} subgraph of $\sigma$ containing the vertices
$v_0$ and $v_2$, i.e. the stable $A$-graph with vertices $v_0$ and
$v_2$ and one flag attached to each vertex corresponding to the marked
point $q$ and $\phi q$ respectively.  There is an associated
$1$-morphism $\Kbm{}{\PP^n,\sigma} \rightarrow \Kbm{}{\PP^n,\tau'}$.
Because $C_1$ is not contracted by $f$, this $1$-morphism is smooth.
In particular, the invariant of $T_{[f]} \Kbm{}{\PP^n,\sigma}$ is at
least as large as the invariant of $T_{[f]} \Kbm{}{\PP^n,\tau'}$.  Of
course $\Kbm{}{\PP^n,\tau'}$ is simply a product of the two factors from
$v_0$ and $v_2$, $\Kbm{0,1}{\PP^n,e_1} \times \Kbm{0,1}{\PP^n,e_1}$,
where $e_1$ is the degree of $f|_{D_q}:D_q \rightarrow \PP^n$.  The
Zariski tangent space is then a direct sum of the two
factors from $v_0$ and $v_2$.  The automorphism $\phi$ permutes
the two factors and $\phi^2$ acts as an automorphism of each factor.

\medskip\noindent
Let $\gamma$ be a generator for the character group of $\Gamma =
\langle \phi \rangle$.  Then also $\gamma$ is a generator for the
character group of $\langle \phi^2 \rangle$.  The rank of $T
\Kbm{0,1}{\PP^n,e_1}$ equals $(n+1)e_1 + (n-3) + 1$.  Consider the
invariant $\alpha'_\gamma$ of this $\langle \phi^2
\rangle$-representation with respect to $\gamma$.  One contribution
comes from the marked point $q$; this contribution is positive.  
So the invariant
is positive.  By Lemma~\ref{lem-induced}, the invariant of $T
\Kbm{}{\PP^n,\tau'}$ as a $\Gamma$-representation with respect to
$\gamma$ is,
\begin{equation}
\alpha_\gamma( T \Kbm{}{\PP^n,\tau'}) \geq \alpha'_\gamma + \lt(
(n+1)e_1 + (n-3) + 1 \rt)\frac{1}{2}.
\end{equation}
The right-hand-side of the equation is a minimum when $n=1$ and $e_1 =
1$, in which case it is still larger than $\frac{1}{2}$ (since
$\alpha'_\gamma$ is positive).  So the invariant of
$\Kbm{}{\PP^n,\sigma}$ is larger than $\frac{1}{2}$.  And the
invariant of $\lt( T'_q \otimes T''_q \rt) \oplus \lt( T'_{\phi q}
\otimes T''_{\phi q} \rt)$ is larger than $\frac{1}{2}$.  Therefore
the invariant of $T_{[f]} \Kbm{0,0}{\PP^n,e}$ is larger than $1$.
\end{proof}

\medskip\noindent
Next consider the coarse moduli space of $C_d$.  As in
Section~\ref{sec-proof}, let $(\varrho:M \rightarrow \PP^1, \iota: Y
\times \PP^1 \rightarrow M, B_y \rightarrow M, \PP(N\oplus \mathbf{1})
\rightarrow M)$ denote the deformation to the normal cone associated
to the inclusion $Y \hookrightarrow B$, where $B=\Kbm{0,0}{\PP^n,e}$.
Let $\widetilde{\pi}_d: \widetilde{C}_d \rightarrow M^o$ denote the
projective Abelian cone.

\medskip\noindent
\begin{lem} \label{lem-can} 
If $e\geq 3$ and if $d+ e \leq n$, then the map to the coarse moduli
space
\begin{equation}
\widetilde{C}_d \times_{M^o} N \rightarrow (\widetilde{C}_d
\times_{M^o} N )_{\text{coarse}} 
\end{equation}
is an isomorphism away from codimension $2$, and the coarse moduli
space $( \widetilde{C}_d \times_{M^o} N )_{\text{coarse}}$ is a
normal, $\QQ$-Gorenstein variety with only canonical singularities.
\end{lem}

\medskip\noindent
\begin{proof}
Of course $\widetilde{C}_d \times_{M^o} N$ is normal and Gorenstein,
therefore the coarse moduli space is normal and $\QQ$-Gorenstein.  To
see that the coarse moduli map is an isomorphism away from codimension
$2$ and that the coarse moduli space is canonical, recall the
resolution of the Deligne-Mumford stack, $\widetilde{C}_d \times_{M^o}
N$. 
The resolution is constructed as
follows.  First of all, the projection $N \rightarrow Y$ is
$M^{(0)}(Y,\mc{R}^\vee, \text{pr}_{\mathbb{G}}^* T)$.  By
Proposition~\ref{prop-smooth}, there is a $1$-morphism of stacks
$u^{e-1,0}:M^{(e-1)} \rightarrow N$ such that $M^{(e-1)} \rightarrow
Y$ is representable and smooth.  There is a projective bundle $C_d'$
over $M^{(e-1)}$ and a morphism $C_d' \rightarrow \widetilde{C}_d
\times_{M^o} N$ that is a resolution of singularities.  Observe that
also $C_d' \rightarrow Y$ is representable and smooth.  By
Corollary~\ref{cor-mults}, the relative canonical divisor of $C_d'
\rightarrow \widetilde{C}_d \times_{M^o} N$ is effective.

\medskip\noindent
Now consider the coarse moduli spaces $Y \rightarrow
Y_{\text{coarse}}$, $C_d' \rightarrow C_{d,\text{coarse}}'$ and
$\widetilde{C}_d \times_{M^o} N \rightarrow (\widetilde{C}_d
\times_{M^o} N)_\text{coarse}$.  By Proposition~\ref{prop-RSBT2}, the
morphism $Y \rightarrow Y_{\text{coarse}}$ is an isomorphism away from
codimension $2$ and $Y_{\text{coarse}}$ is canonical (this corresponds
to the case $(e,1)$ where $e\geq 3$).  By
Corollary~\ref{cor-DMsmooth}, also $C_d' \rightarrow
C_{d,\text{coarse}}'$ is an isomorphism away from codimension $2$ and
$C_{d,\text{coarse}}'$ is canonical.  There is an open substack $U
\subset \widetilde{C}_d \times_{M^o} N$ such that $C_d' \rightarrow
\widetilde{C}_d \times_{M^o} N$ is an isomorphism over $U$ and such
that the complement of $U$ has codimension at least $2$.  And the
morphism $U \rightarrow U_{\text{coarse}}$ is an isomorphism away from
codimension $2$.  Therefore also the morphism
\begin{equation}
\widetilde{C}_d \times_{M^o} N \rightarrow (\widetilde{C}_d
\times_{M^o} N )_{\text{coarse}} 
\end{equation}
is an isomorphism away from codimension $2$.  

\medskip\noindent
Because $C_d' \rightarrow C_{d,\text{coarse}}'$ is an isomorphism away
from codimension $2$, the relative canonical divisor of
$C_{d,\text{coarse}}' \rightarrow (\widetilde{C}_d \times_{M^o}
N)_\text{coarse}$ equals the image of the canonical divisor of $C_d'
\rightarrow \widetilde{C}_d \times_{M^o} N$.  Therefore the relative
canonical divisor is effective.  But also $C_{d,\text{coarse}}'$ is
canonical.  It follows that also $(\widetilde{C}_d \times_{M^o}
N)_\text{coarse}$ is canonical.  This finishes the proof.
\end{proof}

\medskip\noindent
\begin{rmk} \label{rmk-can} 
If $e=2$ and $d + 3 \leq n$, 
the second part of the lemma also holds by a
slightly more \emph{ad hoc} argument
(note that this inequality is worse than
the inequality $d+ e \leq n$).  In this case $Y$ is a $\ZZ/2\ZZ$-gerbe
over its coarse moduli space.  And $N = M^{(0)}$ is a vector bundle of
$1\times (n-1)$-matrices.  So the only stratum to blow up to form
$M^{(1)}$ is the zero section.  Doing this, the
$\ZZ/2\ZZ$-invariant locus of $M^{(1)}$ is the whole exceptional
divisor $E$.  A simple computation shows that the $1$-morphism $C_d'
\rightarrow M^{(1)}$ preserves all stabilizer groups of geometric
points (i.e. the induced homomorphisms of stabilizer groups are
isomorphisms).  Therefore the morphism $C_d' \rightarrow
C_{d,\text{coarse}}'$ is a morphism to a smooth variety ramified of
ramification index $1$ along the preimage of $E$, i.e. it does not
satisfy the first part of the lemma.  However, it is straightforward
to compute that the relative canonical divisor of
$C_{d,\text{coarse}}' \rightarrow (\widetilde{C}_d \times_{M^o}
N)_\text{coarse}$ is $\frac{n-3-d}{2} E_{\text{coarse}}$.  Therefore
$(\widetilde{C}_d \times_{M^o} N)_\text{coarse}$ is canonical when $d+
3 \leq n$.
\end{rmk}  

\section{Conjectures about $\kbm{0,0}{X,e}$} \label{sec-conj}

\medskip\noindent
\begin{conj}[Inversion of Adjunction, Conj. 7.3 ~\cite{KollarPairs}]
  \label{conj-ioa} 
Let $X$ be a normal variety, $S$ a normal Cartier divisor and $B=\sum
b_i B_i$ a $\QQ$-divisor.  Assume that $K_X + S + B$ is $\QQ$-Cartier.
Then
\begin{equation}
\text{totaldiscrep}(S, B|S) = \text{discrep}(\text{Center} \cap S \neq
\emptyset, X, S+B),
\end{equation}
where the notation on the right means that we compute the discrepancy
using only those divisors whose center on $X$ intersects $S$.
\end{conj}

\medskip\noindent
\begin{conj} \label{conj-1} 
If $e \geq 3$ and $d+e \leq n$, the coarse moduli space
$C_{d,\text{coarse}}$ is a normal, $\QQ$-Gorenstein variety with only
canonical singularities.  If $e=2$ and $d+3 \leq n$, the coarse moduli
space $C_{d,\text{coarse}}$ is a normal, $\QQ$-Gorenstein variety with
only canonical singularities.
\end{conj}

\medskip\noindent
\begin{conj} \label{conj-2} 
If $e \geq 3$, $d+e \leq n$, and if $X\subset \PP^n$ is a general
hypersurface of degree $d$, then $\kbm{0,0}{X,e}$ is a normal,
$\QQ$-Gorenstein variety with only canonical singularities.  If $e=2$,
$d+3 \leq n$ and if $X \subset \PP^n$ is a general hypersurface of
degree $d$, then $\kbm{0,0}{X,2}$ is a normal, $\QQ$-Gorenstein
variety with only canonical singularities.
\end{conj}

\medskip\noindent
\begin{prop} \label{prop-ioac1} 
Suppose that $d+e \leq n$.
\begin{enumerate}
\item[(i)]
If $e \geq 2$ then the coarse moduli space $( C_d)_{\text{coarse}}$ is
normal, $\QQ$-Gorenstein and Kawamata log terminal.
\item[(ii)]
If $e \geq 3$ and $d+e \leq n$, then the coarse moduli map
\begin{equation}
C_d \rightarrow ( C_d )_{\text{coarse}}
\end{equation}
is an isomorphism away from codimension $2$. 
\item[(iii)]
Conjecture~\ref{conj-ioa} implies Conjecture~\ref{conj-1}.
\end{enumerate}
\end{prop}

\medskip\noindent
\begin{proof}
\textbf{(i)}:
First of all, the stack $C_d$ is normal and Gorenstein with canonical
singularities by Theorem~\ref{thm-main}.  So the coarse moduli space
$C_{d,\text{coarse}}$ is normal and $\QQ$-Gorenstein.  And by
~\cite[Prop. 3.16]{KollarPairs}, $C_{d,\text{coarse}}$ is Kawamata log
terminal.  

\medskip\noindent
\textbf{(ii)}:
Denote by $Z\subset C_d$ the closed substack where the map $C_d
\rightarrow (C_d )_{\text{coarse}}$ is not an isomorphism.  Denote by
$\widetilde{Z} \subset \widetilde{C}_d$ the closed substack where the
map $\widetilde{C}_d \rightarrow ( \widetilde{C}_d )_{\text{coarse}}$
is not an isomorphism.  Clearly $\widetilde{Z} \cap \rho^{-1}(\AAA^1)
= Z \times \AAA^1$.  Of course $Z$ is invariant under the action of
$\text{GL}_{n+1}$.  By the same argument as in the proof of
Theorem~\ref{thm-main}, every irreducible component of $\widetilde{Z}$
has non-empty intersection with the fiber over $\infty$,
i.e. $\widetilde{C}_d \times_{M^o} N$.  By Lemma~\ref{lem-can}, if
$e\geq 3$, then every irreducible component of $\widetilde{Z} \cap
(\widetilde{C}_d \times_{M^o} N)$ has codimension at least $2$.  By
Krull's Hauptidealsatz, every irreducible component
of $\widetilde{Z}$ has codimension at least $2$.  Therefore every
irreducible component of $Z \subset C_d$ has codimension at least $2$.

\medskip\noindent
\textbf{(iii)}:
As in the proof of
Theorem~\ref{thm-main}, let $W \subset \kbm{0,0}{\PP^n,e}$ be the
largest open substack over which $C_{d,\text{coarse}}$ is canonical.
This is a $\text{GL}_{n+1}$-invariant open set, so to prove that $W$
is all of $\kbm{0,0}{\PP^n,e}$, it suffices to prove that $W$ contains
the image of $Y$.

\medskip\noindent
Let $\widetilde{C}_d \rightarrow M^o$ be as in
Section~\ref{sec-proof}.  Let $M^o \rightarrow M^o_{\text{coarse}}$
and $\widetilde{C}_d \rightarrow \widetilde{C}_{d,\text{coarse}}$ be
the coarse moduli spaces.  Let $W'\subset M^o_{\text{coarse}}$ be the
largest open subset over which $\widetilde{C}_{d,\text{coarse}}$ is
canonical.  Of course $W' \cap \varrho^{-1}(\AAA^1) = W \times
\AAA^1$.  Now by Lemma~\ref{lem-can}, the Cartier divisor
$(\widetilde{C}_d \times_{M^o} N)_{\text{coarse}}$ in
$\widetilde{C}_{d,\text{coarse}}$ is normal and canonical.  Assuming
Conjecture~\ref{conj-ioa} is true, there is an open
subvariety of $\widetilde{C}_{d,\text{coarse}}$ containing
$(\widetilde{C}_d \times_{M^o} N)_{\text{coarse}}$ that is canonical,
i.e. $W'$ contains $\varrho^{-1}(\infty)$.  By the same argument as in
the proof of Theorem~\ref{thm-main}, $W$ contains the
image of $Y$, i.e. $W$ is all of $\kbm{0,0}{\PP^n,e}$.  So
Conjecture~\ref{conj-1} is true.
\end{proof}

\medskip\noindent
\begin{prop} \label{prop-c1c2} 
Let $d, e$ and $n$ be positive integers such that $d+e \leq n$.  
Let $X \subset \PP^n$ be a general hypersurface of
degree $d$.
\begin{enumerate}
\item[(i)]
If $e \geq 2$, the coarse moduli space $\kbm{0,0}{X,e}$ is normal,
$\QQ$-Gorenstein and Kawamata log terminal.
\item[(ii)]
If $e \geq 3$, the coarse moduli map
\begin{equation}
\Kbm{0,0}{X,e} \rightarrow \kbm{0,0}{X,e}
\end{equation}
is an isomorphism away from codimension $2$.
\item[(iii)]
Conjecture~\ref{conj-1} implies Conjecture~\ref{conj-2}.
\end{enumerate}
\end{prop}

\medskip\noindent
\begin{proof}
This is the same argument as in the proof of Corollary~\ref{cor-main}.
\end{proof}

\medskip\noindent
\begin{rmk} \label{rmk-c1c2} 
In (ii), if $e=2$ then the coarse moduli map fails to be an
isomorphism precisely on the locus $Y \cap \Kbm{0,0}{X,2}$.  By direct
computation, for $X$ general this locus has dimension $2n-d-1$.  And
$\Kbm{0,0}{X,2}$ has dimension $3n-d-2$.  Therefore, if $d +3\leq n$,
then the coarse moduli map $\Kbm{0,0}{X,2} \rightarrow \kbm{0,0}{X,2}$
is an isomorphism away from codimension $2$.
\end{rmk}

\section{The canonical class on $\Kbm{0,r}{X,e}$} \label{sec-canKBM}

\medskip\noindent 
Let $X\subset \PP^n$ be a complete intersection of $c$ hypersurfaces
of degrees $\mathbf{d} = (d_1,\dots,d_c)$.  Associated to the
inclusion morphism, there is a $1$-morphism of Kontsevich moduli
spaces $\Kbm{0,r}{X,e} \rightarrow \Kbm{0,r}{\PP^n,e}$.  This
$1$-morphism is representable and is a closed immersion.  The image is
the zero locus of a section $\sigma$ of a locally free sheaf
$\mc{P}_{\mathbf{d}}$ in the small \'etale site of
$\Kbm{0,r}{\PP^n,e}$.

\medskip\noindent
If $\sigma$ is a \emph{regular} section, 
the dualizing sheaf $\omega'$ on $\Kbm{0,r}{X,e}$ can be expressed 
as the pullback from
$\Kbm{0,r}{\PP^n,e}$ of the tensor product $\omega\otimes
\text{det}(\mc{P}_{\mathbf{d}})$, where $\omega$ is the dualizing sheaf
on $\Kbm{0,r}{\PP^n,e}$.  Pandharipande computed the $\QQ$-Picard
group of $\Kbm{0,r}{\PP^n,e}$ in ~\cite{QDiv}.  And he computed
the $\QQ$-divisor class of $\omega$ in ~\cite{Pand97}.  The purpose of
this section is to compute the class $\text{det}(\mc{P}_{\mathbf{d}})$
in terms of the standard generators of the $\QQ$-Picard group, and
thereby compute the $\QQ$-divisor class of $\omega'$ in the case that
$\sigma$ is a regular section.

\medskip\noindent
Let $p\colon \mathcal{C}\rightarrow \Kbm{0,r}{\PP^n,e}$ denote the
universal curve.  Let $f\colon \mathcal{C} \rightarrow \PP^n$ denote
the universal morphism.  For each integer $d$, form the
pullback sheaf $f^*\OO_{\PP^r}(d)$.  Define $\mathcal{E}_d$ to be
the pushforward $p_*(f^*\OO_{\PP^r}(d)$.  More generally, given an
ordered sequence $\mathbf{d} = (d_1,\dots,d_c)$ of integers, define
$\OO_{\PP^n}(\mathbf{d})$ to be $\OO_{\PP^n}(d_1)\oplus \dots \oplus
\OO_{\PP^n}(d_c)$ and define $\mc{P}_{\mathbf{d}}$ to be
$\mc{P}_{d_1}\oplus \dots \oplus \mc{P}_{d_c}$.

\medskip\noindent
There is a pullback map on global sections,
\begin{equation}
H^0\lt( \PP^n, \OO_{\PP^n}(d) \rt) \rightarrow H^0\lt( \mathcal{C},
f^*\OO_{\PP^n}(d) \rt) \rightarrow H^0\lt( \Kbm{0,r}{\PP^n,e},
\mc{P}_d \rt).
\end{equation}
Denote the composite map by $f^*$.  More generally, given an
ordered sequence $\mathbf{d}$, there is a pullback map on global
sections,
\begin{equation}
f^*\colon H^0\lt( \PP^n, \OO_{\PP^n}(\mathbf{d}) \rt) \rightarrow
H^0\lt( \Kbm{0,r}{\PP^n,e}, \mc{P}_{\mathbf{d}} \rt).
\end{equation}

\medskip\noindent
\begin{lem} \label{applem1} 
If $d\geq 0$, then $\mathcal{P}_d$ is a locally free sheaf of rank
$de+1$ in the small \'etale site of $\Kbm{0,r}{\PP^n,e}$ and
$R^ip_*\lt( f^*\OO_{\PP^n}(d) \rt)$ is zero for $i>0$.  More
generally, if $\textbf{d} = (d_1,\dots,d_c)$ and $d_1,\dots,d_c \geq
0$, then $\mc{P}_{\textbf{d}}$ is a locally free sheaf of rank
$|\textbf{d}|e+c$, where $|\mathbf{d}| = d_1 + \dots + d_c$ and
$R^ip_* \lt( f^*\OO_{\PP^n}(\textbf{d}) \rt)$ is zero for $i>0$.
\end{lem}

\medskip\noindent
\begin{proof}
This has been proved in other places, in particular it is proved as
part of the the proof of ~\cite[Lemma 4.5]{HRS2}.
\end{proof}

\medskip\noindent 
Now let $d_1,\dots,d_c$ be a sequence of positive integers, and let
$s=(s_1,\dots, s_c)$ be a global section of $H^0\lt( \PP^n,
\OO_{\PP^n}(\mathbf{d}) \rt)$.  Let $X\subset \PP^n$ be the zero locus
of $s$.  Let $\sigma$ denote the pullback section $f^*s\in H^0\lt(
\Kbm{0,r}{\PP^n,e}, \mc{P}_{\mathbf{d}}\rt).$

\medskip\noindent
\begin{lem}\label{applem2} 
The zero locus of $\sigma$ as a closed substack of
$\Kbm{0,r}{\PP^n,e}$ is the image of the closed immersion
$\Kbm{0,r}{X,e} \rightarrow \Kbm{0,r}{\PP^n,e}$.
\end{lem}

\medskip\noindent
\begin{proof}
This is also proved as part of the proof of ~\cite[Lemma 4.5]{HRS2}.
\end{proof}

\medskip\noindent
Since $\mc{P}_{\mathbf{d}}$ is a locally free sheaf of rank
$e|\mathbf{d}| + c$, and since $\Kbm{0,r}{\PP^n,e}$ is smooth, 
$\sigma$ is a regular section iff the codimension of
$\Kbm{0,r}{X,e}$ in $\Kbm{0,r}{\PP^n,e}$ equals $e|\mathbf{d}|+c$.  In
this case, the generalized version of the adjunction
theorem proves the dualizing sheaf $\omega'$ on $\Kbm{0,r}{X,e}$ is the
pullback of the sheaf $\omega\otimes \text{det}(\mc{P}_{\mathbf{d}})$,
where $\omega$ is the dualizing sheaf on $\Kbm{0,r}{\PP^n,e}$.

\medskip\noindent
In ~\cite{QDiv}, Pandharipande described the $\QQ$-Picard group
$\text{Pic}\lt(\Kbm{0,r}{\PP^n,e}\rt) \otimes \QQ$.  For simplicity,
assume that $n>1$ and $e>0$ and also that $(n,e) \neq (2,2)$.  The
divisor class $\mathcal{H}$ is defined as the image of a positive
generator $h^2\in \text{CH}^2(\PP^n)$ under the composition
\begin{equation}
\text{CH}^2(\PP^n) \xrightarrow{f^*} \text{CH}^2(\mathcal{C})\otimes
\QQ \xrightarrow{p_*} \text{CH}^1(\Kbm{0,r}{\PP^n,e})\otimes \QQ.
\end{equation}
For each of the $r$ sections $g_i:\Kbm{0,r}{\PP^n,e} \rightarrow
\mathcal{C}$, the divisor class $\mathcal{L}_i$ is defined as the
image of a positive generator $h\in \text{CH}^1(\PP^n)$ under pullback
by $f\circ g_i$.  Finally, for each \emph{weighted partition}
$P=(A\cup B,e_A,e_B)$ of $(\{1,\dots,r\},e)$, there is the class
$\Delta_P$ of the corresponding boundary stratum of
$\Kbm{0,r}{\PP^n,e}$.  A weighted partition is a datum consisting of a
partition $A\cup B$
of $\{1,\dots, r\}$, and of a pair of nonnegative integers $e_A, e_B$ such
that $e_A + e_B = e$, satisfying the condition $|A| \geq 2$ (resp. $|B|
\geq 2$) if $e_A = 0$ (resp. $e_B = 0$).  The corresponding boundary
stratum is the closure of the locally closed substack parametrizing
stable maps whose dual graph is of type $(A\cup B, e_A, e_B)$.
Pandharipande's result is that the $\QQ$-Picard group is a
$\QQ$-vector space with basis,
\begin{equation}
\{\mathcal{H}\} \cup \{ \mathcal{L}_i | i=1,\dots, r\} \cup \{
\Delta_P | P = (A\cup B, e_A, e_B) \}.
\end{equation}
In the case that $r=0$, for $i=0,\dots,\lt[\frac{e}{2}\rt]$, denote
by $\mathcal{D}_{i,0}$ the $\QQ$-divisor class $\Delta_{P}$ where $P
=(\emptyset \cup \emptyset, i, e-i)$.  And for $r>0$, for
$i=0,\dots,\lt[ \frac{e}{2}\rt]$ and $j=0,\dots, r$ denote by
$W_{i,j}$ the set of weighted partitions $\{(A\cup B, e_A,e_B)|
|A| = j, e_A = i\}$.  Denote by $\mathcal{D}_{i,j}$ the
$\QQ$-divisor class
\begin{equation}
\mathcal{D}_{i,j} = \sum_{P\in W_{i,j}} \Delta_P.
\end{equation}

\medskip\noindent
In ~\cite{Pand97}, Pandharipande computed the $\QQ$-divisor class of
the dualizing sheaf $\omega$ in terms of the basis above.

\medskip\noindent
\begin{prop}[Pandharipande, Prop. 2 ~\cite{Pand97}] \label{Pand1}
The dualizing sheaf $\omega$ on $\Kbm{0,0}{\PP^n,e}$ has $\QQ$-divisor
class,
\begin{equation}
\omega = \frac{1}{2e}\lt[ -(n+1)(e+1)\mathcal{H}+
\sum_{i=1}^{\lt[ \frac{e}{2}\rt] } \lt(( n+1)(e-i)i - 4e\rt)
\mathcal{D}_{i,0} \rt].
\end{equation}
\end{prop}

\medskip\noindent
\begin{prop}[Pandharipande, Prop. 3 ~\cite{Pand97}] \label{Pand2}
The first Chern class of the dualizing sheaf $\omega$ on
$\Kbm{0,r}{\PP^n,e}$ has $\QQ$-divisor class,
\begin{eqnarray*}
C_1(\omega) = \frac{1}{2e^2}\lt[ -(n+1)(e+1)e + 2r\rt] \mathcal{H}
-\frac{1}{2e} \sum_{p=1}^{n} \mathcal{L}_p + \\
\frac{1}{2e^2}\sum_{i=0}^{\lt[\frac{e}{2}\rt]} \sum_{j=0}^{r} \lt[
(n+1)e(e-i)i + 2e^2j - 4eij + 2ri^2 - 4e^2 \rt] \mathcal{D}_{i,j}.
\end{eqnarray*}
\end{prop}

\medskip\noindent
It remains to compute the $\QQ$-divisor class of the first Chern class
$C_1(\mc{P}_{\textbf{d}})$.  First compute for each integer
$d\geq 0$, the first Chern class $C_1(\mc{P}_d)$.  This computation is
an application of
the Grothendieck-Riemann-Roch formula ~\cite[Thm. 15.2]{F}.
Observe that $p$ is a representable morphism between smooth
Deligne-Mumford stacks with projective coarse moduli space.  So one
can deduce Grothendieck-Riemann-Roch for $p$ from
Grothendieck-Riemann-Roch for the coarse moduli spaces using
~\cite{Vis}.  Alternatively, one can use the Grothendieck-Riemann-Roch
theorem of Toen ~\cite{Toen}.

\medskip\noindent
By Lemma~\ref{applem1}, the element in $K$-theory,
$Rp_{!}[f^*\OO_{\PP^n}(d)]$ equals $[\mc{P}_d]$.  So, by the
Grothendieck-Riemann-Roch formula, 
\begin{equation}
\text{ch}[\mc{P}_d] = p_*\lt( f^*\text{ch}[\OO_{\PP^n}(d)]\cap
\text{todd}(p) \rt).
\end{equation}

\medskip\noindent
Denote by $h\in \text{CH}^1(\PP^n)$ the first Chern class of
$\OO_{\PP^n}(1)$.  Then, up to terms in $\text{CH}^3(\PP^n)$, 
\begin{equation}
\text{ch}[\OO_{\PP^n}(d)] = 1 + dh + \frac{d^2}{2}h^2 + \dots
\end{equation}
By ~\cite[Section 3.E]{HarrMorr}, up to terms in
$\text{CH}^3(\mathcal{C})\otimes \QQ$, 
\begin{equation}
\text{todd}(p) = 1 - \frac{1}{2}C_1(\omega_p) + \frac{1}{12}\lt( \eta
+ C_1(\omega_p)^2 \rt) + \dots, 
\end{equation}
where $\eta$ is the $\QQ$-divisor class of the ramification locus of
$p$.  By ~\cite[Lemma 2.1.2]{QDiv}, $p_*\lt( \eta + C_1(\omega_p)^2
\rt)$ equals zero.  Therefore, up to terms in
$\text{CH}^2(\Kbm{0,r}{\PP^n,e})\otimes \QQ$, 
\begin{eqnarray*}
 p_*\lt( f^*\text{ch}[\OO_{\PP^n}(d)]\cap \text{todd}(p) \rt) = \ \ \
 \ \ \ \\
p_*\lt(df^*(h) - \frac{1}{2}C_1(\omega_p) \rt) +
\frac{d^2}{2}p_*f^*(h^2) -\frac{d}{2} p_*\lt(f^*(h)\cap C_1(\omega_p)
\rt).
\end{eqnarray*}
Clearly $p_*(f^*(h))$ is just $e$ and $p_*(C_1(\omega_p))$ is just
$-2$.  By definition, $p_*f^*(h^2)$ is the divisor class
$\mathcal{H}$.  

\medskip\noindent
\begin{lem}\label{appChernpre} 
In the $\QQ$-Picard group of $\Kbm{0,r}{\PP^n,e}$, 
\begin{equation}
p_* \lt( f^*(h)\cap C_1(\omega_p) \rt) = \frac{1}{d}\lt[ -\mathcal{H}
+ \sum_{i=1}^{\lt[ \frac{e}{2} \rt]}\sum_{j=0}^{r} (e-i)i
\mathcal{D}_{i,j} \rt].
\end{equation}
\end{lem}

\medskip\noindent
\begin{proof}
Denote by $\alpha$ the difference of the right-hand-side of the
equation from the left-hand-side.  The proposition is
that $\alpha$ equals $0$ in the $\QQ$-Picard group.

\medskip\noindent
The method of proof is the same as in ~\cite[Section 1.2]{QDiv}.
Consider the class $\mathcal{S}$ of all pairs $(B,\zeta)$ where $B$ is
a smooth complete curve, $\zeta:B \rightarrow \Kbm{0,r}{\PP^n,e}$ is a
$1$-morphism and such that $(B,\zeta)$ satisfies,
\begin{enumerate}
\item[(i)] 
for the pullback of the universal curve, $p_\zeta:\mathcal{C}_\zeta
\rightarrow B$, $\mathcal{C}_\zeta$ is a smooth surface,
\item[(ii)] 
the general fiber of $p_\zeta$ is a smooth, rational curve,
\item[(iii)] 
every singular fiber of $p_\zeta$ has exactly two irreducible
components, and
\item[(iv)] 
blowing down one irreducible component in each singular fiber yields a
ruled surface over $B$.
\end{enumerate}
In ~\cite{QDiv}, it is proved that for any nonzero divisor class
$\beta$ in the $\QQ$-Picard group of $\Kbm{0,r}{\PP^n,e}$, there is a
pair $(B,\zeta)$ in $\mathcal{S}$ such that $\zeta^*(\beta)$ has
nonzero degree on $B$.  So to prove the proposition, it suffices to
prove that for every pair $(B,\zeta)$ in $\mathcal{S}$,
$\zeta^*(\alpha)$ has degree zero.

\medskip\noindent
Suppose $(B,\zeta)$ is in $\mathcal{S}$.  Let $(E_1\cup
E'_1,\dots,E_m\cup E'_m)$ denote the irreducible components of the
singular fibers of $p_\zeta$.  Let $s:B\rightarrow \mathcal{C}_\zeta$
denote a section of $p_\zeta$ not intersecting any of
$E_1,\dots, E_m$ (by (iv), such a section exists), and let $S$
denote $s(B)$.  Let $F$ denote any smooth fiber of $p_\zeta$.  In the
group of numerical equivalence classes, $N^1(\mathcal{C}_\zeta)$, the
classes $[S],[F], [E_1],\dots, [E_m]$ give a basis for
$N^1(\mathcal{C}_\zeta)$ as a free $\ZZ$-module.

\medskip\noindent
Denote $k = -\text{deg}([S]\cap [S])$.  By straightforward computation,
\begin{equation}
\lt\{ \begin{array}{ll}
\text{deg}([F]\cap [F])     & = \ 0 \\ 
\text{deg}([F]\cap [S])     & = \ 1 \\
\text{deg}([F]\cap [E_i])   & = \ 0 \\
\text{deg}([S]\cap [E_i])   & = \ 0 \\
\text{deg}([E_i]\cap [E_i]) & = -1 \\
\text{deg}([E_i]\cap [E_j]) & = \ 0,\ \  i\neq j
\end{array} \rt.
\end{equation}
By the adjunction formula, $\lt( \zeta^*\omega_p \otimes
\OO_{\mathcal{C}_\zeta}(E_i) \rt)|_{E_i} \cong \omega_{E_i}$.
Therefore, $\text{deg}\lt( \zeta^*C_1(\omega_p) + [E_i]\rt) \cap [E_i]
= -2$, i.e. $\text{deg}(\zeta^*C_1(\omega_p)\cap [E_i]) = -1$.  
Similarly, $\lt( \zeta^*\omega_p \otimes
\OO_{\mathcal{C}_\zeta}(F) \rt)|_{F} \cong \omega_F$.  Therefore
$\text{deg}\lt( \zeta^*C_1(\omega_p) + [F] \rt)\cap [F] = -2$,
i.e. $\text{deg}( \zeta^*C_1(\omega_p) \cap [F]) = -2$.  Finally, by
adjunction $\lt( \zeta^*\omega_p \otimes
\OO_{\mathcal{C}_\zeta}(S) \rt)|_S$ is isomorphic to the relative
dualizing sheaf of $p_\zeta|_{S}:S \rightarrow B$.  But this is an
isomorphism, so the relative dualizing sheaf is just $\OO_S$.
Therefore $\text{deg}\lt( \zeta^*C_1(\omega_p) + [S]\rt) \cap
[S] = 0$, i.e. $\text{deg}( \zeta^*C_1(\omega_p) \cap [S]) = k$.
Putting this all together, the numerical equivalence class
of $\zeta^*C_1(\omega_p)$ equals
\begin{equation}
\zeta^* C_1(\omega_p) = -2[S] -k[F] + \sum_{i=1}^m [E_i].
\end{equation}

\medskip\noindent
Denote $l = \text{deg}(\zeta^*f^*(h))\cap [S]$.  For each
$i=1,\dots, m$, define $e_i = \text{deg}(\zeta^*f^*(h)\cap [E_i])$.
By a similar computation as above, the numerical
equivalence class of $\zeta^* f^*(h)$ equals
\begin{equation}
\zeta^* f^*(h) = e[S] + (l + ek)[F] - \sum_{i=1}^m e_i[E_i].
\end{equation}
Thus, $\text{deg}(\zeta^*C_1(\omega_p)\cap \zeta^*f^*(h))
= -2l -ek + \sum_{i=1}^m e_i$.  Similarly,
$\text{deg}(\zeta^*f^*(h) \cap \zeta^*f^*(h)) = 2el+e^2k -\sum_{i=1}^m
ee_i + \sum_{i=1}^m e_i(e-e_i)$,
i.e. $-e\text{deg}(\zeta^*C_1(\omega_p) \cap \zeta^*f^*(h)) +
\sum_{i=1}^m e_i(e-e_i)$.  Finally, observe that, 
\begin{equation}
\text{deg}\lt( \zeta^* \sum_{i=1}^{\lt[\frac{e}{2}\rt]} \sum_{j=0}^{r}
i(e-i) \mathcal{D}_{i,j} \rt) = \sum_{i=0}^m e_i(e-e_i).
\end{equation}
In conclusion,
\begin{equation}
\text{deg}\zeta^* p_*\lt(C_1(\omega_p) \cap f^*(h) \rt) =
-\frac{1}{e}\text{deg}(\zeta^* \mathcal{H}) + \frac{1}{e}
\sum_{i=1}^{\lt[\frac{e}{2} \rt]} \sum_{j=0}^r i(e-i)
\text{deg}(\zeta^* \mathcal{D}_{i,j}),
\end{equation}
just as required.
\end{proof}

\medskip\noindent
\begin{prop}\label{appChern1} 
On $\Kbm{0,r}{\PP^n,e}$, the $\QQ$-divisor class of the first Chern
class of $\mc{P}_d = p_*(f^*\OO_{\PP^n}(d))$ equals,
\begin{equation}
C_1(\mc{P}_d) = \frac{d}{2e}\lt[ (ed+1)\mathcal{H} -
\sum_{i=1}^{\lt[\frac{e}{2}\rt]} \sum_{j=0}^{r}
i(e-i)\mathcal{D}_{i,j} \rt].
\end{equation}
More generally, for $\textbf{d} = (d_1,\dots,d_c)$, the $\QQ$-divisor
class of the first Chern class of $\mc{P}_{\textbf{d}} =
p_*(f^*\OO_{\PP^n}(\textbf{d}))$ equals,
\begin{equation}
C_1(\mc{P}_{\textbf{d}}) = \frac{1}{2e}
\lt( \prod_{k=1}^c (ed_k + 1) \rt) \lt[
\lt(\sum_{k=1}^c d_k\rt) \mathcal{H} + \lt(\sum_{k=1}^c
\frac{d_k}{ed_k+1} \rt) \sum_{i=1}^{\lt[\frac{e}{2}\rt]} \sum_{j=0}^c
i(e-i) \mathcal{D}_{i,j} \rt].
\end{equation}
\end{prop}

\medskip\noindent
\begin{proof}
Substituting the result from Lemma~\ref{appChernpre} into the
Grothendieck-Riemann-Roch formula yields, up to terms in
$\text{CH}^2(\Kbm{0,r}{\PP^n,e}) \otimes \QQ$,
\begin{equation}
\text{ch}[\mc{P}_d] = (ed+1) + \frac{d^2}{2}\mathcal{H} +
\frac{d}{2e} \lt( \mathcal{H} - \sum_{i=1}^{\lt[ \frac{e}{2} \rt]}
\sum_{j=0}^r i(e-i) \mathcal{D}_{i,j} \rt) + \dots
\end{equation}
Since $\text{ch}[\mc{P}_d] = \text{rank}(\mc{P}_d) + C_1(\mc{P}_d) +
\dots$, the first part of the proposition follows.

\medskip\noindent
Since $\mc{P}_{\mathbf{d}} \cong \oplus_{k=1}^c \mc{P}_{d_i}$, we have
the formula
\begin{equation}
C_1(\mc{P}_{\mathbf{d}}) = \lt( \prod_{k=1}^c
\text{rank}(\mc{P}_{d_k}) \rt) \sum_{k=1}^c
\frac{C_1(\mc{P}_{d_k})}{\text{rank}(\mc{P}_{d_k})}. 
\end{equation}
Substituting the first part of the proposition gives the second part
of the proposition.
\end{proof}

\medskip\noindent
The following corollaries follow immediately from Proposition
~\ref{appChern1}.  They are stated separately for
notational convenience.

\medskip\noindent
Let $s\in H^0(\PP^n,\OO_{\PP^n}(d))$ be a section with zero locus
$X\subset \PP^n$.  Consider the locally free sheaf $\mc{P}_d$ on
$\Kbm{0,0}{\PP^n,e}$.  Denote by $\sigma = f^*s$ the induced section
of $\mc{P}_d$; the closed substack $\Kbm{0,0}{X,e} \subset
\Kbm{0,0}{\PP^n,e}$ is the zero locus of $\sigma$.

\begin{cor}\label{appCherncor1}
If the section $\sigma = f^*s$ of
$\mc{P}_d$ is a regular section, i.e. if $\Kbm{0,0}{X,e}$ has the
expected codimension $ed+1$ in $\Kbm{0,0}{\PP^n,e}$,  then the
$\QQ$-divisor class of the first Chern class of the dualizing sheaf on
$\Kbm{0,0}{X,e}$ equals,
\begin{equation}
\frac{1}{2e}\lt[ \lt( (d^2 - n -1)e - (n+1-d) \rt) \mathcal{H} +
\sum_{i=1}^{\lt[\frac{e}{2} \rt]} \lt((n+1-d)i(e-i) - 4e\rt)
\mathcal{D}_{i,0} \rt].
\end{equation}
\end{cor} 

\medskip\noindent
Let $\mathbf{d} = (d_1,\dots,d_c)$ be a sequence of positive
integers.  Let $s\in H^0(\PP^n, \OO_{\PP^n}(\mathbf{d}))$ be a section
with zero locus $X \subset \PP^n$.  Consider the locally free sheaf
$\mc{P}_{\mathbf{d}}$ on $\Kbm{0,r}{\PP^n,e}$.  Denote by $\sigma =
f^*s$ the induced section; the closed substack $\Kbm{0,0}{X,e} \subset
\Kbm{0,0}{\PP^n,e}$ is the zero locus of $\sigma$.

\medskip\noindent
\begin{cor}\label{appCherncor2} 
If the section $\sigma = f^*s$ is a regular section,
i.e. $\Kbm{0,r}{X,e}$ has the expected codimension $e|\mathbf{d}|
+ c$ in $\Kbm{0,r}{\PP^n,e}$,  then the $\QQ$-divisor class of the
first Chern class of the dualizing sheaf on $\Kbm{0,r}{X,e}$ equals,
\begin{eqnarray*}
\frac{1}{2e^2}\lt[e \lt( \prod_{k=1}^c ed_k+1 \rt) \lt( \sum_{k=1}^c
d_k \rt) + 2r - (n+1)(e+1) \rt] \mathcal{H} -\frac{2}{e} \sum_{j=1}^r
\mathcal{L}_j + \\ 
 \sum_{j=0}^r
 j\mathcal{D}_{0,j} + 
\frac{1}{2e^2}\sum_{i=1}^{\lt[ \frac{e}{2} \rt]} \sum_{j=0}^r
 \lt[ \lt(\prod_{k=1}^c ed_k +1\rt)\lt(\sum_{k=1}^c \frac{ed_k}{ed_k+1} \rt)
i(e-i) + \rt. \\ 
 \lt. (n+1)ei(e-i) + 2e^2j - 4eij + 2ri^2 - 4e^2 \rt] \mathcal{D}_{i,j}
\end{eqnarray*} 
\end{cor}

\medskip\noindent
\begin{lem}\label{applembigB} 
Let $X\subset \PP^n$ be a projective scheme.
\begin{enumerate}
\item[(i)] 
If every geometric generic point of $\Kbm{0,0}{X,e}$ parametrizes a
stable map mapping birationally to its image, then the pullback of
the $\QQ$-divisor class $\mathcal{H}$ in the $\QQ$-Picard group of
$\Kbm{0,0}{X,e}$ is big.  Moreover, the pullback of this divisor class
to the seminormalization of $\Kbm{0,0}{X,e}$ is Cartier and
base-point-free.
\item[(ii)]
If every geometric generic point of $\Kbm{0,0}{X,e}$ parametrizes a
an $a$-normal smooth rational curve in $X$, then the pullback of
the $\QQ$-divisor class $C_1(\mc{P}_a)$ is an effective Cartier
divisor.
\item[(iii)]
If every geometric generic point of $\Kbm{0,0}{X,e}$ parametrizes a
stable map with irreducible domain, then for $i=1,\dots, \lt[
\frac{e}{2} \rt]$, the pullback of $\mathcal{D}_{i,0}$ is an effective
$\QQ$-Cartier divisor.
\end{enumerate}
\end{lem}

\medskip\noindent
\begin{proof}
\textbf{(i)}:
To prove (i), replace $\Kbm{0,0}{X,e}$ by its
seminormalization $\Kbm{0,0}{X,e}_{\text{sn}}$.  Consider the
universal curve $p:\mc{C} \rightarrow \Kbm{0,0}{X,e}_{\text{sn}}$ and
the universal morphism $f:\mc{C} \rightarrow X$.  Form the closed
image subscheme $C \subset \Kbm{0,0}{X,e}_{\text{sn}} \times X$ of
$(p,f)$.  Now $C$ is a \emph{well defined family of algebraic cycles}
in the sense of ~\cite[Defn. I.3.10]{K}.  By ~\cite[Thm. I.3.21]{K},
there is a Chow variety $\text{Chow}_{1,e}(X)$ and an induced morphism
$\Kbm{0,0}{X,e}_{\text{sn}} \rightarrow \text{Chow}_{1,e}(X)$, the
\emph{Kontsevich-Chow morphism}.  By the
construction in ~\cite[Section I.3.23]{K}, there is an ample
invertible sheaf on $\text{Chow}_{1,e}(X)$ such that $\mc{H}$ is the
pullback of this ample invertible sheaf.  By the hypothesis
that every geometric generic point of $\Kbm{0,0}{X,e}$ parametrizes a
stable map that maps birationally to its image, the Kontsevich-Chow
morphism is
generically finite.  Therefore $\mc{H}$ is base-point-free and big.

\medskip\noindent
\textbf{(ii)}:
Let $W\subset H^0(\PP^n, \OO_{\PP^n}(a))$ be a general vector subspace
of dimension $ae+1$.  There is an induced map $W \otimes_\CC
\OO_{\Kbm{0,0}{X,e}} \rightarrow \mc{E}_a$ and the first Chern class
is simply the locus where this map fails to be an isomorphism.  By the
assumption that every geometric generic point of $\Kbm{0,0}{X,e}$
parametrizes an $a$-normal stable map which, there is no irreducible
component of $\Kbm{0,0}{X,e}$ contained in this locus.
Therefore this locus is an effective Cartier divisor.

\medskip\noindent
\textbf{(iii)}:
Finally, by construction the boundary divisors are effective
$\QQ$-Cartier divisors on $\Kbm{0,0}{\PP^n,e}$.  If no irreducible
component of $\Kbm{0,0}{X,e}$ is contained in the boundary, then the
pullbacks of the boundary divisors are effective $\QQ$-Cartier
divisors on $\Kbm{0,0}{X,e}$.
\end{proof}

\medskip\noindent
\begin{cor} \label{cor-Kbig} 
Let $X\subset \PP^n$ be a general hypersurface of degree $d$.
\begin{enumerate}
\item[(i)]
If $d < \text{min}(n-3,\frac{n+1}{2})$ and $d^2 \geq n+2$, then for
$e>>0$ the canonical divisor of $\Kbm{0,0}{X,e}$ is big.
\item[(ii)]
If $d < \text{min}(n-6,\frac{n+1}{2})$ and $d^2 + d \geq 2n + 2$, then
for every $e>0$ the canonical divisor of $\Kbm{0,0}{X,e}$ is big.
\end{enumerate}
In particular, if also $e \geq 3$ and $d+e \leq n$ or $e=2$ and $d+3
\leq n$, then Conjecture~\ref{conj-2} implies that $\kbm{0,0}{X,e}$ is
of general type.
\end{cor}

\medskip\noindent
\begin{proof}
When $d<\frac{n+1}{2}$, then ~\cite[Prop. 7.4]{HRS2} implies
that $\Kbm{0,0}{X,e}$ satisfies the hypotheses of (i) and (iii)
of Lemma~\ref{applembigB}.  Combining this with the formula from
Corollary~\ref{appCherncor1} gives (i) and (ii).

\medskip\noindent 
Finally, by Proposition~\ref{prop-c1c2} and Remark~\ref{rmk-c1c2},
if $e\geq 3$ and $d+e \leq n$ or if $e=2$ and $d+3 \leq n$, then the
coarse moduli map $\Kbm{0,0}{X,e} \rightarrow \kbm{0,0}{X,e}$ is an
isomorphism away from codimension $2$ so that the canonical bundle of
$\kbm{0,0}{X,e}$ equals the canonical bundle of $\Kbm{0,0}{X,e}$.
Therefore Conjecture~\ref{conj-2} implies that $\kbm{0,0}{X,e}$ is of
general type.
\end{proof}

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