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\begin{document}

\title[Rational curves on hypersurfaces]{Rational curves on
hypersurfaces of low degree} 

\author[J. Harris]{Joe Harris} 
 \address{Department of Mathematics \\ 
   Harvard University \\ 
   Cambridge, MA 02138, USA}
 \email{harris@math.harvard.edu} 

\author[M. Roth]{Mike Roth}
 \address{Department of Mathematics and Statistics \\ 
  Queens University \\ 
  Kingston, Ontario, K7L3N6, Canada} 
 \email{mikeroth@mast.queensu.ca}
 
\author[J. Starr]{Jason Starr}
 \thanks{The third author was partially
   supported by an NSF Graduate Research Fellowship and a Sloan
   Dissertation Fellowship} 
 \address{Department of Mathematics \\ 
   Massachusetts Institute of Technology \\ 
   Cambridge, MA 02139, USA}
\email{jstarr@math.mit.edu} 

\date{\today}


\begin{abstract}
  We prove that for $n>2$ and $d<\frac{n+1}{2}$, a \emph{general}
  complex hypersurface $X\subset \PP^n$ of degree $d$ has the property
  that for each integer $e$ the scheme $R_e(X)$ parametrizing degree
  $e$, smooth rational curves on $X$ is an integral, local complete
  intersection scheme of ``expected'' dimension $(n+1-d)e+(n-4)$.

The techniques used in the proof include:
\begin{enumerate}
\item  Classical results about lines on hypersurfaces including a new
result about flatness of the projection map from the space of
\emph{pointed lines}.
\item  The Kontsevich moduli space of stable maps,
$\Kgnb{0}{r}{X}{e}$.  In particular we use the
deformation theory of stable maps, properness of the stack
$\Kgnb{0}{r}{X}{e}$, and the decomposition of
$\Kgnb{0}{r}{X}{e}$ described in ~\cite{BM}.
\item  A version of Mori's bend-and-break lemma.
\end{enumerate}
\end{abstract}

\maketitle

\section{Summary}~\label{sec-sum}

\subsection{Brief Summary}~\label{subsec-brsum}
All schemes we consider will be $\CC$-schemes and all morphisms will
be morphisms of $\CC$-schemes.  All (absolute) products will be over
$\CC$.

For a projective scheme $X$ over $\CC$ along with an ample line bundle
$L$ we define $R_e(X)$ to be the open subscheme of the Hilbert scheme
$\text{Hilb}^{et+1}(X/k)$ which parametrizes smooth rational curves of
degree $e$ lying in $X$.

\begin{thm}~\label{thm-main}  Let $n > 2$ be an integer and let $d$ be a
  positive integer such that $d < \frac{n+1}{2}$.  For a general
  hypersurface $X\subset \PP^n$ of degree d and for every integer
  $e\geq 1$, the scheme $R_e(X)$ is an integral, local complete
  intersection scheme of dimension $(n+1-d)e+(n-4)$.
\end{thm}

The idea of the proof is as follows.  There is an embedding of
$R_e(X)$ into the smooth scheme $R_e(\PP^n)$. Denote by
$\pi:U_e(\PP^n)\rightarrow R_e(\PP^n)$ the universal family of
rational curves in $\PP^n$ and by $\rho:U_e(\PP^n)\rightarrow \PP^n$
the \emph{evaluation} morphism.  Then $R_e(X)$ is the scheme of zeroes
of a section of the locally free sheaf $\pi_* \rho^* \OO_{\PP^n}(d)$.
Thus to prove that $R_e(X)$ is a local complete intersection scheme,
it suffices to prove that the codimension of $R_e(X)$ in $R_e(\PP^n)$
equals the rank of $\pi_* \rho^* \OO_{\PP^n}(d)$.

The remainder of the proof is a ``deformation and specialization''
argument: we embed the non-proper scheme $R_e(X)$ as an open subscheme
of a proper scheme which is still modular, i.e. we choose a ``modular
compactification''. Then we show that every generic point in $R_e(X)$
specializes to a point in the ``boundary'' of the compactification.
We use deformation theory to study the irreducible components of the
boundary of the compactification.  In particular we show that a
general point of each irreducible component of the boundary is a
unibranch point of the compactification whose local ring is reduced
and has the expected dimension.  This reduces the proof to a
combinatorial argument.

\subsection{Detailed Summary}~\label{subsec-detsum}

In the next few paragraphs we will give a detailed summary of the
proof.  Our compactification consists of the embedding of $R_e(X)$ as
an open subscheme in the Kontsevich moduli space $\Kgnb{0}{0}{X}{e}$
parametrizing stable maps to $X$.  We recall the partition of
$\Kgnb{0}{0}{X}{e}$ into locally closed subsets defined in~\cite{BM};
we call this partition the \emph{Behrend-Manin decomposition} (our
partition differs slightly from that in~\cite{BM}).  In particular,
the image of $R_e(X)$ is a dense open subset of a component of this
partition. We identify certain \emph{basic components} as those
components of the partition parametrizing stable maps such that each
irreducible component of the domain curve is mapped to a line in $X$.

We prove a new result about lines on $X$.  We define the incidence
correspondence of \emph{pointed lines} in X:
\begin{equation}
F_{0,1}(X)=\{(p,l) |p \text{ a point}, l \text{ a line}, p\in
l\subset X\}.
\end{equation}
We prove that for a general hypersurface $X\subset \PP^n$ of degree
$d\leq n-1$, the projection morphism $F_{0,1}(X)\rightarrow X$ is flat
of relative dimension $n-d-1$.  From this theorem it easily follows
that each basic component $B$ is an integral scheme whose general
point is a unibranch point of $\Kgnb{0}{0}{X}{e}$ at which
$\Kgnb{0}{0}{X}{e}$ is reduced of dimension $(n+1-d)e+(n-4)$. Thus for
each basic component $B$ there is a unique irreducible component
$M(B)$ of $\Kgnb{0}{0}{X}{e}$ which contains $B$, and $M(B)$ is
reduced and has dimension $(n+1-d)e+(n-4)$.

Using a version of the bend-and-break lemma of Mori, we prove that
every irreducible component of $\Kgnb{0}{0}{X}{e}$ is of the form
$M(B)$ for some basic component $B$.  Using this fact and results
about flatness, we bootstrap to prove that each evaluation map
$\Kgnb{0}{r}{X}{e}\rightarrow X$ is flat of the expected dimension and
is \emph{generically unobstructed}.  This implies that each
$\Kgnb{0}{r}{X}{e}$ (including $r=0$) has the expected dimension and
is generically smooth.  Thus $\Kgnb{0}{r}{X}{e}$ is a reduced, local
complete intersection stack of the expected dimension and it only
remains to prove that $\Kgnb{0}{r}{X}{e}$ is irreducible, i.e. it
remains to prove that all of the irreducible components $M(B)$ are
actually equal.

To prove that all of the irreducible components $M(B)$ are equal, we
observe that there is a combinatorially defined equivalence relation
defined on the set of basic components $B$ such that equivalent basic
components, $B\cong B'$ satisfy $M(B)=M(B')$.  Thus we are reduced to
a combinatorial argument which proves that all basic components are
equivalent.

Along the way we generalize the strategy of proof above so that it
could apply to smooth projective schemes $X$ other than hypersurfaces
$X\subset \PP^n$ of degree $d< \frac{n+1}{2}$ (this is made completely
explicit for complete intersections in $\PP^n$).  One is reduced to
proving
\begin{enumerate}
\item The evaluation morphism $\Kgnb{0}{1}{X}{e}\rightarrow X$ is
  flat, generically unobstructed and the general fiber is
  geometrically irreducible.
\item For each positive integer $e$ at most the \emph{threshold
    degree} of $X$, the evaluation morphism
    $\Kgnb{0}{1}{X}{e}\rightarrow X$ is flat of the expected
    dimension.
\item For each positive integer $e$ at most the threshold degree of
    $X$, the stack $\Kgnb{0}{0}{X}{e}$ is irreducible.
\end{enumerate}
The most difficult condition to verify seems to be $(2)$, but it is
our hope that this can be verified for a larger class of Fano schemes
$X$ than the hypersurfaces above.

In ~\cite{KP}, Kim and Pandharipande proved irreducibility and rationality 
of the stacks $\Kgnb{0}{r}{X}{\beta}$ when $X$ is a homogeneous
variety for a linear algebraic group (and $\beta$ is a numerical
equivalence class of curves on
$X$).  
In particular, when $X$ is a
linear or quadric hypersurface in $\PP^n$, with $n\geq 4$, it follows
from ~\cite[Corollary 1]{KP} that $\Kgnb{0}{r}{X}{e}$ is
irreducible and from ~\cite[Theorem 3]{KP} that
$\Kgnb{0}{r}{X}{e}$ is rational.  
This paper can be seen as a generalization of the
irreducibilty result ~\cite[Corollary 1]{KP} to hypersurfaces
$X\subset \PP^n$ of degree roughly $d\leq\frac{n}{2}$. In a
forthcoming paper, \cite{HS2}, we give a partial generalization of the
rationality result of~\cite[Theorem 3]{KP} to hypersurfaces $X\subset
\PP^n$ of degree roughly $d\leq \sqrt{n}$.

\subsection{Notation}~\label{subsec-not}
Given a $\CC$-vector space $W$, $\PP W$ denotes the projective space
$\text{Proj}(\oplus_{d\geq 0} S^d(W^*))$ which parametrizes
one-dimensional linear subspaces of $W$ (not one-dimensional quotient
spaces of $W$).  Given any integers $k\leq n$, $G(k,n)$ denotes the
Grassmannian which parametrizes $k$-dimensional linear subspaces of
$\CC^n$.  For a triple of integers $k\leq l\leq n$, $F((k,l),n)$
denotes the partial flag variety which parametrizes partial flags
$V_1\subset V_2\subset \CC^n$ of linear subspaces with
$\text{dim}(V_1)=k$ and $\text{dim}(V_2)=l$.


\subsection{Acknowledgments}~\label{subsec-acknow}  
The authors would like to thank Tom Graber, Ravi Vakil, Andreas
Gathmann, Olivier Debarre, and especially Johan de Jong for many
useful discussions.

\section{Lines on Hypersurfaces}~\label{sec-lines}

We denote $W=H^0(\PP^n,\OO_{\PP^n}(d))$ so that $\PP W$ parametrizes
degree $d$ hypersurfaces $X\subset \PP^n$.  Let $\cX \subset \PP W
\times \PP^n$ denote the universal family of degree $d$ hypersurfaces
in $\PP^n$.  For each degree $d$ hypersurface $X\subset \PP^n$, denote
by $F_1(X)$ the subscheme of $G(2,n+1)$ which parametrizes lines
$L\subset X\subset \PP^n$.  Denote by $F((1,2),n+1)$ the partial flag
variety parametrizing pairs $(p,L)$ where $L\subset \PP^n$ is a line
and $p\in L$ is a point.  Denote by $F_{0,1}(X)$ the subscheme of
$F((1,2),n+1)$ which parametrizes pairs $(p,L)$ with $p\in L \subset
X\subset \PP^n$.  

Similarly, let $F_1(\cX)\subset \PP W\times
G(2,n+1)$ denote the subscheme parametrizing pairs $(X,L)$ with $L\in
F_1(X)$, and let $F_{0,1}(\cX)\subset \PP W\times F((1,2),n+1)$ denote
the subscheme parametrizing triples $(X,p,L)$ with $(p,L)\in
F_{0,1}(X)$.  There are projection morphisms
\begin{eqnarray}
\pi_0:\PP W \times F((1,2),n+1)\rightarrow \PP W, \\
\pi_1:\PP W \times F((1,2),n+1)\rightarrow \PP^n, \\
\pi_2:\PP W \times F((1,2),n+1)\rightarrow G(2,n+1)
\end{eqnarray}
By construction, the morphism $(\pi_0,\pi_1):F_{0,1}(\cX)\rightarrow
\PP W\times \PP^n$ factors through $\cX \subset \PP W\times \PP^n$.
Denote by $\rho:F_{0,1}(\cX)\rightarrow \cX$ the induced morphism.
For a particular hypersurface $X\in \PP W$, denote by
$\rho_X:F_{0,1}(X)\rightarrow X$ the fiber of $\rho$.

The main result of this section is the following theorem:
\begin{thm} \label{thm-thm1}  Let $d$ be a positive integer with $d \leq
n-1$.  For a 
general hypersurface $X\in \PP W$ the morphism
$\rho_X:F_{0,1}(X)\rightarrow X$ is flat of relative dimension
$n-d-1$.
\end{thm}

We give the proof in the remainder of this section.  From now on
we assume that $d$ is given such that $d\leq n-1$.

Denote by $\OO$ the structure sheaf of $\PP W \times F((1,2),n+1)$ and
denote by $\OO_F$ the $\OO$-module which is the pushforward of the
structure sheaf of $F_{0,1}(\cX)$.  On $G(2,n+1)$ we have a universal
rank 2 subbundle of $\OO_{G(2,n+1)}^{\oplus n+1}$.  Denote by $S$ the
pullback under $\pi_2$ of this universal subbundle to $\PP W\times
F((1,2),n+1)$.  And denote by $U$ the pullback under $\pi_0$ of the
universal rank 1 subbundle $\OO_{\PP W}(-1) \subset
H^0(\PP^n,\OO_{\PP^n}(d))\otimes_\CC \OO_{\PP W}$.  By restricting a
section of $H^0(\PP^n,\OO_{\PP^n}(d))$ to a line $L\subset \PP^n$
parametrized by a point $[L]\in G(2,n+1)$, we get a map $U\rightarrow
\text{Sym}^d(S^\vee)$.  By adjunction, this gives rise to a map
$\text{Sym}^d(S)\otimes_\OO U \rightarrow \OO$, whose image is exactly the
ideal sheaf of $F_{0,1}(\cX)$.  In other words, there is a partial
resolution of coherent sheaves:
\begin{equation}
\begin{CD}
  \text{Sym}^d(S)\otimes_{\OO} U @> \sigma >> \OO @>>> \OO_F @>>> 0.
\end{CD}
\end{equation}
In other words, $F_{0,1}(\cX)$ is the zero scheme of the global
section $\OO\rightarrow \text{Sym}^d(S^\vee)\otimes_\OO U^\vee$ which is
the transpose of $\sigma$.

Since $\text{Sym}^d(S)\otimes_\OO U$ is locally free of rank $d+1$, every
irreducible component of $\cX$ has codimension at most $d+1$ in $\PP
W\times F((1,2),n+1)$.  Therefore, every (nonempty) fiber of $\rho$
has dimension at least $n-d-1$.  We define ${\mc U}\subset \cX$ as a
set to be
\begin{equation}
{\mc U} = \lt\{ (X,p)\in \cX |\ \text{dim}(\rho^{-1}(X,p))\leq n-d-1
\rt\}.
\end{equation}
It follows by upper semicontinuity of the fiber dimension that ${\mc
  U}$ is a Zariski open subset of $\cX$, and we give it the
corresponding structure of open subscheme of $\cX$.  A priori ${\mc
  U}$ might contain points $(X,p)\subset \cX$ for which
$\rho^{-1}(X,p)$ is empty.  But by ~\cite[exercise V.4.6]{K}, it
follows that $\rho$ is surjective (also this exercise rederives the
statement above about dimensions of fibers).

Notice that the projection morphism $\pi_0:\cX\rightarrow \PP^n$ is a
projective bundle whose fiber over $p\in \PP^n$ is identified, as a
subscheme of $\PP W$, with the hyperplane parametrizing $X\in \PP W$
with $p\in X$.  In particular, $\cX$ is a smooth $k$-scheme.  Given
the map $\sigma$ above, we can form the Koszul complex of locally free
$\OO$-modules in the usual way.  By ~\cite[Theorem 17.4 (iii)(4)]{Ma},
this complex is acyclic over ${\mc U}$.  Therefore the fibers of
$\rho$ over ${\mc U}$, considered as subschemes of the appropriate
fiber of $\pi_1:F((1,2),n+1)\rightarrow \PP^n$, all have equal Hilbert
polynomial.  Since ${\mc U}$ is smooth, it follows from ~\cite[Theorem
III.9.9]{H} that $\rho$ is flat over ${\mc U}$.

Let ${\mc Y} \subset F_{0,1}(\cX)$ denote the complement of ${\mc U}$
with the induced, reduced scheme structure.  Theorem 2 is equivalent
to the statement that $\pi_0|_{\mc Y}:{\mc Y}\rightarrow \PP W$ is not
surjective.  Denote by $e$ the codimension of $\mathcal{Y}$ in
$\mathcal{X}$.  Since the fiber dimension of $\mathcal{X}\rightarrow
\PP W$ is $n-1$, to prove that $\mathcal{Y}$ fails to dominate $\PP
W$, it suffices to prove that $e > n-1$.  In the remainder of this
section we will prove that $e>n-1$.

On $\PP^n$ let $Q$ denote the locally free quotient sheaf of
$\OO_{\PP^n}^{\oplus n+1}$ by $\OO_{\PP V}(-1)$.  The dual injection
$Q^\vee \hookrightarrow \lt(\OO_{\PP^n}^{\oplus n+1}\rt)^\vee$ can be
considered as a filtration of $\lt(\OO_{\PP^n}^{\oplus n+1}\rt)^\vee$.
The $d$th symmetric product of this filtration is a filtration of $W
\otimes_\CC \OO_{\PP^n}$:
\begin{equation}
  W \otimes_\CC \OO = F^{0,d} \supset F^{1,d}
\supset \dots \supset F^{d,d} \supset F^{d+1,d} = 0.
\end{equation}
Here $F^{i,d}$ is the locally free subsheaf of $ W \otimes_\CC \OO$
which is the image of the multiplication map
\begin{equation}
\text{Sym}^{d-i} \lt(\OO_{\PP^n}^{\oplus n+1}\rt)^\vee  \otimes_{\OO_{\PP^n}} 
\text{Sym}^i Q^\vee\rightarrow W\otimes_\CC \OO.
\end{equation}
The associated graded sheaves of this filtration $G^{i,d} =
F^{i,d}/F^{i+1,d}$ are canonically isomorphic to the sheaves
$\OO_{\PP^n}(d-i)\otimes_{\OO_{\PP^n}} \text{Sym}^i Q^\vee$.

In particular, notice that $F^{1,d}$ is simply the kernel of the
evaluation map $W\otimes_\CC \OO_{\PP^n}\rightarrow \OO_{\PP^n}(d)$,
i.e. the vector bundle parametrizes pairs $(\Phi,p)$ where $\Phi\in W$
is such that $\Phi(p)=0$.  We identify a nonzero section $\Phi\in W$,
up to nonzero scaling, with the hypersurface it defines $X=V(\Phi)$.
Then the associated projective bundle $\PP F^{1,d}$ inside $\PP
W\times \PP^n$ is the closed subscheme parametrizing pairs $(X,p)$
with $p\in X$, i.e. $\PP F^{1,d} = \cX$.  To prove the inequality
$e>n-1$ from above, it suffices to prove that for each $p\in \PP^n$ 
(equivalently for any $p\in \PP^n$ by homogeneity) the intersection
${\mc Y}\cap \pi_0^{-1}(p) \subset \cX$ has codimension greater than
$n-1$ in $\pi_0^{-1}(p)$.  In the remainder of this section we will
prove this.

Observe the filtration above is not split on $\PP^n$.  But we can find
a covering of $\PP^n$ by open affine subschemes $A_\alpha \subset
\PP^n$ over which we do have a splitting (for example, the standard
covering by complements of coordinate hyperplanes).  Here by
\emph{splitting} we mean an isomorphism of bundles over $A_\alpha$
\begin{equation}
\begin{CD} s: W \otimes_\CC \OO_{A_\alpha} @>>>
  \oplus_{j=0}^{d} \OO_{\PP^n}(d-j)\otimes_{\OO_{\PP^n}} \text{Sym}^j
  Q^\vee|_{A_\alpha}
\end{CD}
\end{equation}
which maps $F^{i,d}|_{A_\alpha}$ to the subbundle $\oplus_{j=i}^{d}
\OO_{\PP^n}(d-j)\otimes_{\OO_{\PP^n}} \text{Sym}^j Q^\vee|_{A_\alpha}$
and such that the induced isomorphism $G^{i,d}|_{A_\alpha}\rightarrow
\OO_{\PP^n}(d-i) \otimes_{\OO_{\PP^n}} \text{Sym}^i
Q^\vee|_{A_\alpha}$ is the isomorphism from above.

Given an open affine $A_\alpha\subset \PP^n$ we can form the
projective bundle $\PP_{A_\alpha}(F^{1,d}|_{A_\alpha})$ over
$A_\alpha$.  Given a splitting $s$ on $A_\alpha$, denote by
\begin{equation}
\Delta_j(s)\subset \PP_{A_\alpha} (F^{1,d}|_{A_\alpha})
\end{equation}
the closed subscheme which parametrizes pairs $(\Phi,x)$, $x\in
A_\alpha, \Phi\in F^{1,d}|x$ such that the $j$th component of $s(\Phi)$
is zero.  Thus $\Delta_0(s)$ is all of $\PP_{A_\alpha}
(F^{1,d}|_{A_\alpha})$.  And, considering $\PP_{A_\alpha}
(F^{1,d}|_{A_\alpha})$ as an open subscheme of $\mathcal{X}$,
$\Delta_1(s)$ is the intersection of $\PP_{A_\alpha}
(F^{1,d}|_{A_\alpha})$ with the singular locus of the projection
morphism $\pi_0:\mathcal{X}\rightarrow \PP W$.  Although $\Delta_0(s)$
and $\Delta_1(s)$ are independent of the choice of $s$, the same is
not true for $\Delta_i(s)$ with $i>1$.  The next result follows
immediately from the definition of the $\Delta_j(s)$.

\begin{lem}\label{lem-lem0}  For $j>0$ the codimension of $\Delta_j(s)$ in
  $\PP_{A_\alpha}(F^{1,d}|_{A_\alpha})$ equals
\begin{equation}
\text{rank } \lt( \OO_{\PP^n}(d-j)\otimes_{\OO_{\PP^n}} \text{Sym}^j Q^\vee
\rt) = \lt( 
\begin{array}{c}  n-1+j \\ n-1 \end{array}
\rt).
\end{equation}
\end{lem}

In particular, for $j>0$ the codimension of $\Delta_j(s)$ in
$\PP_{A_\alpha}(F^{1,d}|_{A_\alpha})$ is at least $n-1+j > n-1$.  For
each open subscheme ${A_\alpha}\subset \PP V$, each splitting $s$, and
each point $p\in {A_\alpha}$, define the locally closed subscheme
\begin{equation}
\mathcal{Y}_{p,s}:=\lt( \mathcal{Y}\cap \pi_0^{-1}(p) \rt)-\cup_{j=1}^d
\lt( \Delta_j(s)\cap \pi_0^{-1}(p) \rt).
\end{equation}
To establish that $e>n-1$, it suffices to prove that the codimension
of ${\mc Y}_{p,s}$ as a subscheme of $\pi_0^{-1}(p)$ 
has codimension greater than $n-1$.  In the remainder of this section
we prove this inequality.

On the complement of the closed subset $\Delta(s) :=\cup_{j=1}^d \lt(
\Delta_j(s) \rt)$ there is a morphism
\begin{equation}
\PP_{A_\alpha} \lt( F^{1,d}|_{A_\alpha} \rt)-\Delta(s) \xrightarrow{\beta}
  \prod_{j=1}^d \PP_{A_\alpha} \lt( \OO_{\PP^n}(d-j)
  \otimes_{\OO_{\PP^n}} \text{Sym}^j Q^\vee \rt)|_{A_\alpha}.
\end{equation}
We identify the space
\begin{equation}
\PP_{A_\alpha} \lt( \OO_{\PP^n}(d-j) \otimes_{\OO_{\PP^n}} 
   \text{Sym}^j Q^\vee \rt)|_{A_\alpha}
\end{equation}  
with the scheme parametrizing degree $j$ hypersurfaces in fibers of
the projection morphism $\PP_{A_\alpha} Q|_{A_\alpha} \rightarrow
{A_\alpha}$.  Thus $\beta$ assigns to each suitable pair $([\Phi],p)$
a sequence of hypersurfaces in $\PP Q|_p$.  Denote this sequence by
$\lt(X_1,\dots,X_j,\dots,X_d\rt)$.  Also, denote by $X=X_1\cap \dots
\cap X_d$ the intersection of these hypersurfaces.

\begin{lem}\label{lem-lem1}  If we denote by $X$ the hypersurface in $\PP V$
  corresponding to $\Phi$, then $X_1\cap \dots\cap X_d$ is the fiber
  of $p\in X$ under $\rho_X:F_{0,1}(X)\rightarrow X$.
\end{lem}

\begin{proof}  
  This is most easily seen by passing to local coordinates.  Let
  $(x_0,\dots,x_n)$ be a system of homogeneous coordinates on $\PP V$
  (i.e. a basis for $V^\vee$) and let $p$ be the point with
  homogeneous coordinates $[0,\dots,0,1]$.  We define a splitting $s$
  as follows: for each degree $d$ homogeneous polynomial $\Phi$ in
  $(x_0,\dots,x_n)$ we have a unique decomposition
\begin{equation}
\Phi = \Phi_d + \Phi_{d-1}x_n+\dots+ \Phi_{d-i}x_n^i +\dots
+\Phi_0 x_n^d
\end{equation}
where each $\Phi_i$ is a homogeneous polynomial of degree $i$ in
$(x_0,\dots,x_{n-1})$.  Then the fiber of $F^{1,d}$ at $p$ consists of
those polynomials such that $\Phi_0=0$ and $\beta(\Phi) =
(\Phi_d,\dots,\Phi_1)$. For any line $L$ passing through $p$ there is
a unique point of the form $y=(a_0,\dots,a_{n-1},0)$ contained in $L$.
Let $\PP^1\rightarrow \PP^n$ be the morphism given by
\begin{equation} 
(t_0,t_1)\mapsto (t_1a_0,t_1a_1,\dots,t_1a_{n-1},t_0+t_1a_n).
\end{equation}  
The image of this morphism is just $L$.  Substituting into $\Phi$
yields the polynomial on $\PP^1$ given by
\begin{eqnarray*}
t_0^d\Phi_d(a_0,\dots,a_{n-1}) + \dots +
t_0^{d-i}t_1^i\Phi_{d-i}(a_0,\dots,a_{n-1})+ \dots \\
+ t_0t_1^{d-1}\Phi_1(a_0,\dots,a_{n-1}).
\end{eqnarray*}   
The line $L$ is contained in $X$ iff this polynomial is identically
zero iff each of the terms $\Phi_i(a_0,\dots,a_n)$ is zero.  One can
show that the homogeneous ideal generated by the terms $\Phi_i$ is
independent of our particular splitting.
\end{proof}

In particular, we conclude that every fiber of $\beta$ which
intersects $\mathcal{Y}$ is contained in $\mathcal{Y}$.  Therefore the
codimension of $\mathcal{Y}_{p,s}$ in $\pi_0^{-1}(p)$ equals the
codimension of the subvariety
\begin{equation}
\beta(\mathcal{Y}) \subset \prod_{j=1}^d
\PP\lt( \OO_{\PP^n}(d-j)\otimes_\OO \text{Sym}^j Q^\vee \rt)|_p.
\end{equation} 
By construction, $\beta(\mathcal{Y})$ is the locus parametrizing
sequences of hypersurfaces in $\PP Q|_p$, $(X_1,\dots,X_d)$, of
degrees $1,\dots, d$ respectively such that the intersection
\begin{equation}
X_{(1,\dots,d)} := X_1\cap \dots \cap X_d
\end{equation}
has dimension greater than $n-d-1$.  So we have reduced
Theorem~\ref{thm-thm1} to the following theorem:

\begin{thm}\label{thm-thm2}  Let $Q$ be a vector space over $k$ of
  dimension $n$ and let $d$ be an integer such that $1\leq d \leq
  n-1$.  Let $\PP_d$ denote the scheme $\prod_{j=1}^d\PP \text{Sym}^j
  Q^\vee$, which parametrizes $d$-tuples $\lt(X_1,\dots,X_d\rt)$ of
  hypersurfaces $X_i\in \PP Q$ of degree $i$.  Denote by $D_d$ the
  closed subscheme of $\PP_d$ which parametrizes sequences
  $\lt(X_1,\dots,X_d\rt)$ such that
\begin{equation}
\text{dim}\lt(X_{(1,\dots,d)}\rt)> n-d-1.
\end{equation}
The codimension of $D_d$ in
$\PP_d$ is greater than $n-1$.
\end{thm}

\begin{proof}
  We will prove this by induction on $d$.  
Consider first the case $d=1$.
Since $D_1=\emptyset$ and
  the dimension of $\PP_d=\PP Q^\vee$ is $n-1$, the result is true for
  $d=1$.

Let $U_d$ denote the open subscheme of $\PP_d$ which is the complement
of $D_d$.  Then for $1\leq d \leq n-2$, $U_{d+1}$ is contained in
$U_d\times \PP \text{Sym}^{d+1} Q^\vee$.  To see this, note that if
$X_{(1,\dots,d)}$ has dimension larger than $n-d-1$, then
$X_{1,\dots,d+1}$ is nonempty and has dimension greater than $n-d-2$:
it is nonempty since $X_{d+1}$ is ample, it has dimension larger than
$n-d-1$ by the Hauptidealsatz.  So we see that the codimension of
$D_{d+1}$ in $\PP_{d+1}$ is the minimum of the codimension of $D_d$ in
$\PP_d$ and the codimension of $D_{d+1}\cap (U_d\times \PP
\text{Sym}^{d+1} Q^\vee)$ in $U_d\times \PP \text{Sym}^{d+1} Q^\vee$.
So by induction we are reduced to showing that the codimension of
$D_{d+1}\cap (U_d\times \PP \text{Sym}^{d+1} Q^\vee)$ in $U_d\times
\PP \text{Sym}^{d+1} Q^\vee$ is larger than $n-1$.

Now suppose that $(X_1,\dots,X_d,X_{d+1})$ is a point in $D_{d+1}\cap
(U_d\times \PP \text{Sym}^{d+1} Q^\vee)$.  By assumption every
irreducible component of $X_{(1,\dots,d)}$ has dimension $n-d-1$.
Since also $X_{(1,\dots,d+1)}$ has dimension $n-d-1$, we conclude that
there is an irreducible component $C\subset X_{(1,\dots,d)}$ such that
$C\subset X_{d+1}$.  If $X_{(1,\dots,d)}=C_1\cup \dots \cup C_r$ is
the irreducible decomposition, then the fiber of $D_{d+1}\cap
(U_d\times \PP\text{Sym}^{d+1} Q^\vee)$ over $(X_1,\dots,X_d)$ (which
we consider as a subscheme of $\PP\text{Sym}^{d+1}Q^\vee$) is just the
union of $i=1,\dots,r$ of the set $B_i\subset \PP\text{Sym}^{d+1}
Q^\vee$ parametrizing hypersurfaces $X_{d+1}$ such that $C_i\subset
X_{d+1}$.  We are reduced to showing that the codimension of each
$B_i$ in $\PP\text{Sym}^{d+1} Q^\vee$ is greater than $n-1$.  We prove
this in a lemma:

\begin{lem}\label{lem-lem2}  Let $Y\subset \PP Q$ be an irreducible
  subscheme such that $\text{dim}Y=n-d-1$.  Let $B(Y)\subset
  \PP\text{Sym}^{d+1} Q^\vee$ be the locus of hypersurfaces $X_{d+1}$
  such that $Y\subset X_{d+1}$.  The codimension of $B(Y)$ is greater
  than $n-1$.
\end{lem}

\begin{proof}  Let $\Lambda\subset \PP Q$ be a $(d-1)$-plane which is disjoint
  from $Y$.  Choose coordinates on $\PP Q$, $(x_0,\dots,x_{n-1})$ with
  respect to which $\Lambda = Z(x_d,\dots,x_{n-1})$.  Let ${\mathbb
    G}_m$ denote the multiplicative group $\SP{\CC[t,t^{-1}]}$.  Let
  $\mu:{\mathbb G}_m\times \PP Q\rightarrow \PP Q$ be the group action
  given by
\begin{equation} 
\mu(t, (x_0, \dots, x_{d-1}, x_d, \dots, x_{n-1})) = (t^{-1}x_0,
\dots, t^{-1}x_{d-1}, tx_d, \dots, tx_{n-1}).
\end{equation}
Since the Hilbert scheme of $\PP Q$ is proper, the valuative criterion
implies that the closed subscheme
\begin{equation}
\mu^{-1}(Y)\subset {\mathbb G}_m\times \PP Q
\end{equation} 
which is flat over ${\mathbb G}_m$, extends over $0$ to yield a closed
subscheme
\begin{equation} 
\mathcal{Y}\subset \AAA^1\times \PP Q
\end{equation} 
which is flat over $\AAA^1$.  It is easy to see that the fiber of
$\mathcal{Y}$ over $0$ is a scheme whose reduced scheme is just
\begin{equation}
Z(x_0,\dots,x_{d-1})\subset \PP Q.
\end{equation}

Now we can form the family
\begin{equation}
\mathcal{B}\subset \AAA^1\times \PP\text{Sym}^{d+1} Q^\vee,
\mathcal{B}_t=B(\mathcal{Y}_t).
\end{equation} 
Over $\mathbb{G}_m$ the fibers of $\mathcal{B}$ are isomorphic.  It
follows by upper semicontinuity that for $t\neq 0$ we have
$\text{dim}(\mathcal{B}_t)\leq \text{dim}(\mathcal{B}_0)$.  And of
course we have
\begin{equation}
\mathcal{B}_0=B(\mathcal{Y}_0)\subset
B \lt(Z(x_0,\dots,x_{d-1}) \rt).
\end{equation}  
So we are reduced to proving the lemma for the special case
$Y=Z(x_0,\dots,x_{d-1})$.  The set B of hypersurfaces $X_{d+1}\subset
\PP Q$ which contain $Z(x_0,\dots,x_{d-1})$ is just the
projectivization of the kernel of the surjective linear map
$$H^0(\PP Q,\OO_{\PP Q}(d+1))\rightarrow H^0(Y,\OO_Y(d+1)).$$  
So the codimension of B in $\PP Q$ equals 
$$\text{dim}_{\CC} H^0(Y,\OO_Y(d+1))=\binom{n}{d+1}.$$  
For $d+1\leq n-1$ (which is one of
our hypotheses) we see that $\binom{n}{d+1}\geq n > n-1$.  We conclude
that the codimension of $B(Y)$ in $\PP\text{Sym}^{d+1}Q^\vee$ is
greater than $n-1$.  This proves the lemma.
\end{proof}

By the above lemma, we conclude that the codimension of each $B_i$ in
$\PP \text{Sym}^{d+1} Q^\vee$ is greater than $n-1$.  So we have
proved Theorem~\ref{thm-thm2}
\end{proof}

Since we had reduced Theorem~\ref{thm-thm1} to Theorem~\ref{thm-thm2},
we have proved Theorem~\ref{thm-thm1}.

While we are discussing results about lines on hypersurfaces, let us
mention two other results about lines on hypersurfaces.

\begin{lem} ~\cite[exercise V.4.4.2]{K} \label{lem-dfmlines}  For
  general $X$ and a general line $L\subset X$, the normal bundle
  $N_{L/X}$ is of the form $\OO_L^{\oplus d-1}\oplus\OO_L(1)^{\oplus
    n-1-d}$.
\end{lem}

\begin{thm} ~\cite[Theorem V.4.3.2]{K} \label{lem-smlines}  For general
  X, the Fano scheme $F_1(X)$ is smooth.  Therefore $F_{0,1}(X)$ is
  smooth.  By generic smoothness, the general fiber of
  $F_{0,1}(X)\rightarrow X$ is smooth.
\end{thm}

\section{Stable $A$-graphs and Stable Maps}\label{sec-stab}

We follow the notation from ~\cite{BM} regarding stable A-graphs.
However, we shall only need to use genus 0 trees.

\subsection{Graphs and Trees}~\label{subsec-graph}

\begin{defn}~\label{def-graph}
  A \emph{graph} $\tau$ is a $4$-tuple
  $(F_\tau,W_\tau,j_\tau,\partial_\tau)$ defined as follows:
\begin{enumerate}
\item $F_\tau$ is a finite set called the set of \emph{flags}
\item $W_\tau$ is a finite set called the set of \emph{vertices}
\item $j_\tau:F_\tau\rightarrow F_\tau$ is an involution
\item $\partial_\tau:F_\tau\rightarrow W_\tau$ is a map called the
\emph{evaluation} map.
\end{enumerate}
In addition we have the auxiliary definitions
\begin{enumerate}
\item the set of \emph{tails} $S_\tau\subset F_\tau$ is the set of
fixed points of $j_\tau$
\item the set of \emph{edges} $E_\tau$ is the quotient of
$F_\tau\setminus S_\tau$ by $j_\tau$
\item for a vertex $v\in W_\tau$, the \emph{valence} of $v$ is defined
to be $\text{val}(v)=\#(\partial^{-1}(v))$.
\end{enumerate}
We shall often write $\text{Flag}(\tau)$ in place of $F_\tau$,
$\text{Vertex}(\tau)$ in place of $W_\tau$, $\text{Tail}(\tau)$ in
place of $S_\tau$, $\text{Edge}(\tau)$ in place of $E_\tau$, and
$\overline{f}$ in place of $j_\tau(f)$.
\end{defn}

We can associate to a graph its \emph{geometric realization} $|\tau|$
which is a CW-complex defined as follows.  The set of
0-cells of $|\tau|$ is
\begin{equation}
|\tau|^0 = \text{Vertex}(\tau) \sqcup
\text{Tail}(\tau).
\end{equation}  
The set of 1-cells of $|\tau|$ is
\begin{equation}
|\tau|^1 = \text{Edge}(\tau)\sqcup \text{Tail}(\tau).
\end{equation}
If $[0,1]$ is a 1-cell associated to an edge
$\lt\{f,\overline{f}\rt\}$, the point 0 is glued to the 0-cell
$\partial f$, and the point 1 is glued to the 0-cell $\partial
\overline{f}$.  If $[0,1]$ is the 1-cell associated to a tail $f$,
the point 0 is glued to the 0-cell $\partial f$, and the point 1 is
glued to the 0-cell $f$.

\begin{defn}~\label{def-tree}
  A \emph{tree} is a connected graph such that $H_1(|\tau|,\ZZ) = 0$,
  i.e. a graph which contains no closed loops.
\end{defn}

One important tree is the \emph{empty tree} $\lambda_\emptyset$, i.e.
the tree such that $\text{Vertex}(\lambda_\emptyset)=\emptyset$.  For
each nonnegative integer $r$ define $\lambda_r$ to be the tree with
one vertex, $\text{Vertex}(\lambda_r)=\{v\}$, and with $r$ flags (all
of which are tails), $\text{Tail}(\lambda_r)=\{f_1,\dots,f_r\}$.
Also, for each pair of nonnegative integers $(r_1,r_2)$, define
$\lambda_{r_1,r_2}$ to be the connected tree with two vertices
$v_1,v_2$, with $r_1$ tails attached to $v_1$ and with $r_2$ tails
attached to $v_2$.

\begin{defn}\label{def-Agraph} 
  An \emph{$A$-graph} is a pair $(\tau,\beta_\tau)$ where $\tau$ is a
  tree and
\begin{equation}
\beta:\text{Vertex}(\tau)\rightarrow \ZZ_{\geq 0}
\end{equation}
is a map called the \emph{$A$-structure}.  We shall often abbreviate
$(\tau,\beta_\tau)$ by just writing $\tau$.  We say that an $A$-graph
$\tau$ is \emph{stable} if for each vertex $v\in\text{Vertex}(\tau)$
such that $\beta_\tau(v)=0$, there are at least 3 distinct flags
$f\in\text{Flag}(\tau)$ such that $\partial f = v$ (i.e. the valence
of v is at least 3).
\end{defn}

One important $A$-graph is the empty graph, $\tau_\emptyset$.  This is
the unique $A$-graph whose underlying graph is $\lambda_\emptyset$.
For each pair of nonnegative integers $r$ and $e$, define $\tau_r(e)$
to be the unique $A$-graph whose underlying graph is $\lambda_r$ and
such that $\beta(v)=e$.  Obviously $\tau_r(e)$ is stable iff either
$r\geq 3$ or $e> 0$.  For each pair of pairs $(r_1,r_2)$ and
$(e_1,e_2)$ where $r_1,r_2,e_1$ and $e_2$ are nonnegative integers,
define $\tau_{r_1,r_2}(e_1,e_2)$ to be the unique $A$-graph whose
underlying graph is $\lambda_{r_1,r_2}$, such that $\beta(v_1)=e_1$
and such that $\beta(v_2)=e_2$.

There is a category whose objects are the stable $A$-graphs.  The
morphisms in this category are each a composition of two basic types
of morphisms: \emph{contractions} and \emph{combinatorial morphisms},
c.f. ~\cite{BM} for the precise definitions.  Essentially a
contraction of $A$-graphs $\phi:\tau\rightarrow \sigma$ is a map from
the vertices of $\tau$ onto the vertices of $\sigma$ which maps
adjacent vertices to adjacent vertices (here two vertices are adjacent
if they are equal or if they are connected by an edge).  And a
combinatorial morphism $\tau \hookleftarrow \sigma$ is the inclusion
of a subgraph $\sigma$ into a graph $\tau$.  The functor which
associates to a stable $A$-graph the corresponding \emph{Behrend-Manin
  stack} is covariant for contractions.  But it is contravariant for
combinatorial morphisms.  Therefore we think of a combinatorial
morphism $\tau \hookleftarrow \sigma$ as a morphism from $\tau$ to
$\sigma$ (which explains our terminology $\tau \hookleftarrow \sigma$
for combinatorial morphisms).

Particularly important are morphisms of graphs which remove tails.
For each stable $A$-graph $\tau$ we define $r_{>0}(\tau)$ to be the
stable $A$-graph obtained by removing every tail
$f\in\text{Tail}(\tau)$ such that $\beta(\partial f) > 0$.  We define
$\tau \hookleftarrow r_{>0}(\tau)$ to be the canonical combinatorial
morphism.  For each stable $A$-graph $\tau$ we define $r_0(\tau)$ to
be the stabilization of the $A$-graph obtained by removing all tails
$f\in\text{Tail}(\tau)$ such that $\beta(\partial f)=0$.  Technically
the canonical morphism of graphs from $\tau$ to $r_0(\tau)$ consists
of both a combinatorial morphism and a contraction.  But we shall
denote it by $\tau \hookleftarrow r_0(\tau)$ just as if it were a
combinatorial morphism.  Finally, we define $r(\tau) :=
r_{>0}(r_0(\tau))=r_0(r_{>0}(\tau))$.

There are numerical invariants associated to an $A$-graph.

\begin{defn}~\label{def-invts}
  Given an $A$-graph $\tau$, define
\begin{equation}
\beta(\tau) = \sum_{v\in \text{Vertex}(\tau)} \beta(v).
\end{equation}
If $(X,L)$ is a polarized variety such that $K_X
\stackrel{\text{num}}{=} m L$ for some integer $m$, define the
\emph{expected dimension}
\begin{equation}
\text{dim}(X,\tau) = -m\beta(\tau) +
\#\text{Tail}(\tau) - \#\text{Edge}(\tau) + (\text{dim}(X)-3).
\end{equation}
\end{defn}

\subsection{Prestable Curves and Dual Graphs}~\label{subsec-dual}

\begin{defn}~\label{def-pcurve}  
  A \emph{prestable curve} with $r$ marked points
  $(C,(x_1,\dots,x_r))$ is a pair where $C$ is a complete, reduced, at
  worst nodal curve and $x_i\in C, i=1,\dots,r$ are distinct,
  nonsingular points of $C$.
\end{defn}

Suppose that $(C,(x_1,\dots,x_r))$ is a connected, prestable curve
whose arithmetic genus is $0$.  One associates to
$(C,(x_1,\dots,x_n))$ a \emph{dual graph}, $\Delta$: a tree whose
vertices $\{v_1,v_2,\dots\}$ correspond to the irreducible components
$\{C_1, C_2,\dots \}$ of $C$, whose edges $\{ \{f_1,\overline{f_1}\},
\{f_2,\overline{f_2}\},\dots \}$ correspond to the nodes
$\{q_1,q_2,\dots \}$ of $C$, and whose tails $\{ g_1,\dots, g_r\}$
correspond to the marked points $\{p_1,\dots,p_r\}$ of $C$.

\begin{defn}~\label{def-pmap} 
  Let $(X,L)$ be a polarized variety.  A \emph{prestable map} is a
  pair
\begin{equation}
((C,(x_1,\dots,x_n)),C\xrightarrow{h} X)
\end{equation}
where $(C,(x_1,\dots,x_n))$ is a prestable curve, and where
$C\xrightarrow{h} X$ is a morphism of $\CC$-schemes.
\end{defn}

Just as one associates to a connected prestable curve
$(C,x_1,\dots,x_n)$ of arithmetic genus $0$ a tree $\Delta(C,x)$, one
can associate an $A$-graph to a prestable map $\lt( (C,x_1,\dots,x_n),
C\xrightarrow{h} X\rt)$ from a connected prestable curve of arithmetic
genus $0$.  The underlying tree of $\Delta(C,x,h)$ is simply
$\Delta(C,x)$.  And, given a component $C_i$ of $C$ with corresponding
vertex $v_i\in \text{Vertex}(\Delta(C,x))$, one defines
\begin{equation}
\beta(v_i)=\int_{C_i} h_i^*(c_1(L)).
\end{equation}
The $A$-graph $\Delta(C,x,h)$ is a \emph{stable} $A$-graph iff
$(C,x,h)$ is a \emph{stable} map.

\subsection{Behrend-Manin stacks}~\label{subsec-BM}
  
We refer the reader to ~\cite{BM} for the definition of the stacks
$\Kbm{X}{\tau}$.  These are proper Deligne-Mumford stacks which
parametrize stable maps along with some extra data.  We shall
sometimes deal with these stacks, but more often we shall deal with
the open substack $\Kbmo{X}{\tau}\subset \Kbm{X}{\tau}$ of
\emph{strict maps} which we now define.

\begin{defn}~\label{defn-strict}
  Let $X$ be a variety, $L$ a line bundle on $X$, and let $\tau$ be a
  stable A-graph.  A \emph{strict $\tau$-map} is a datum
\begin{equation}
((C_v),(h_v:C_v\rightarrow X),(q_f))
\end{equation}
defined as follows:
\begin{enumerate}
\item $(C_v)$ is a set parametrized by $v\in\text{Vertex}(\tau)$ of
smooth rational curves, i.e. each $C_v\cong \PP^1$
\item $(h_v:C_v\rightarrow X)$ is a set parametrized by
$v\in\text{Vertex}(\tau)$ of morphisms of $\CC$-schemes,
\item $(q_f)$ is a set parametrized by $f\in\text{Flag}(\tau)$ of
closed points $q_f\in C_{\partial f}$
\end{enumerate}
and satisfying the following conditions
\begin{enumerate}
\item for $v\in\text{Vertex}(\tau)$, the degree of $h_v^*(L)$ as a
line bundle on $C_v$ is $\beta_\tau(v)$,
\item for $f_1,f_2\in\text{Flag}(\tau)$ distinct flags with $\partial
f_1=\partial f_2$, $q_{f_1}\neq q_{f_2}$,
\item for $f\in\text{Flag}(\tau)$, we have $h_{\partial
f}(q_f)=h_{\partial \overline{f}}(q_{\overline{f}})$.
\end{enumerate}
\end{defn}

\textbf{Convention:} For the empty graph, $\tau_\emptyset$, we define a
strict $\tau_\emptyset$-map to simply be a point in $X$.  Thus the set
of strict $\tau_\emptyset$-maps is simply $X$.

\begin{defn}~\label{defn-strictfam}
  If $T$ is a $\CC$-scheme, then a \emph{family of strict $\tau$-maps}
  over $T$ is a datum
\begin{equation}
((\pi_v:\mathcal{C}_v\rightarrow T),
(h_v:\mathcal{C}_v\rightarrow X), (q_f:T\rightarrow
\mathcal{C}_{\partial f}))
\end{equation}
defined as follows:
\begin{enumerate}
\item $(\pi_v:\mathcal{C}_v\rightarrow T)$ is a set parametrized by
$v\in\text{Vertex}(\tau)$ of smooth, proper morphisms whose geometric
fibers are rational curves
\item $(h_v:\mathcal{C}_v\rightarrow X)$ is a set parametrized by
$v\in\text{Vertex}(\tau)$ of morphisms of $\CC$-schemes
\item $(q_f:T\rightarrow \mathcal{C}_{\partial f})$ is a set parametrized by
$f\in\text{Flag}(\tau)$ of morphisms of schemes such that
$\pi_{\partial f}\circ q_f = id_T$
\end{enumerate}
and satisfying the following conditions
\begin{enumerate}
%\item for $v\in\text{Vertex}(\tau)$, each geometric fiber
%$$\mathcal{C}_v\times_S \text{Spec}\overline{\kappa(s)}$$ is a
%curve of genus $g_\tau(v)$ over $\text{Spec}\overline{\kappa(s)}$
\item for $v\in\text{Vertex}(\tau)$, the degree of $h_v^*(L)$ on each
geometric fiber of $\mathcal{C}_v\rightarrow S$ is $\beta_\tau(v)$
\item for $f_1,f_2\in\text{Flag}(\tau)$ distinct flags with $\partial
f_1 = \partial f_2$, $q_{f_1}$ and $q_{f_2}$ are disjoint sections
\item for $f\in\text{Flag}(\tau)$, we have $h_{\partial f}\circ q_f =
h_{\partial \overline{f}}\circ q_{\overline{f}}$.
\end{enumerate}
\end{defn}

\textbf{Convention:} For the empty graph, $\tau_\emptyset$ we define a
family of strict $\tau_\emptyset$-maps over $T$ to be a morphism
$h:T\rightarrow X$.

Suppose given two families of strict $\tau$-maps over $S$, say
\begin{eqnarray}
\eta= ((\pi_v:\mathcal{C}_v\rightarrow T),
(h_v:\mathcal{C}_v\rightarrow X), (q_f:T\rightarrow
\mathcal{C}_{\partial f})), \\
\zeta=
((\pi'_v:\mathcal{C}'_v\rightarrow T),
(h'_v:\mathcal{C}'_v\rightarrow X), (q'_f:T\rightarrow
\mathcal{C}'_{\partial f})).
\end{eqnarray}

\begin{defn}~\label{defn-strictmor}
  A \emph{morphism} of families of strict $\tau$-maps over $S$,
  $\phi:\eta\rightarrow \zeta$, is a collection of isomorphisms of
  $S$-schemes:
\begin{equation}
\phi =
(\phi_v:\mathcal{C}_v\rightarrow \mathcal{C}'_v)
\end{equation}
indexed by
$v\in\text{Vertex}(\tau)$ and satisfying
\begin{enumerate}
\item for $v\in\text{Vertex}(\tau)$, $h'_v\circ \phi_v = h_v$
\item for $f\in\text{Flag}(\tau)$, $\phi_{\partial f}\circ q_f = q'_f$.
\end{enumerate}
\end{defn}

One defines composition of morphisms in the obvious way.  Notice that
every morphism is an isomorphism.  Thus the category of families of
strict $\tau$-maps over $S$ is a groupoid.  Given a morphism
$S'\xrightarrow{u} S$ and a family $\eta$ of strict $\tau$-maps over
$S$, one has the usual pullback $u^*(\eta)$ which is a family of
strict $\tau$-maps over $S'$.  In this way we have the notion 
a category fibered in groupoids over the category of $\CC$-schemes
along with a \emph{clivage normalis\'ee} in the sense of
~\cite[Exp. VI]{SGA1}.  We denote this category by $\Kbmo{X}{\tau}$.
We will occasionally also denote by $\Kbmo{X}{\tau}$ the associated
lax 2-functor from the category of $\CC$-schemes to the 2-category of
groupoids (with a small skeletal subcategory).  

of a
functor from the category of $\CC$-schemes to the category of
groupoids which associates to each $S$ the groupoid of families of
strict $\tau$-maps over $S$.  We denote this functor by
$\Kbmo{X}{\tau}$.

In every case it is easy to see that $\Kbmo{X}{\tau}$ is a stack in
groupoids over $\CC$.  In many cases this is even a Deligne-Mumford
stack:

\begin{thm}~\label{thm-KBMO}  If $X$ is projective and $L$ is ample,
  the functor $\Kbmo{X}{\tau}$ is a Deligne-Mumford stack which is
  separated and finite type over $\CC$.
\end{thm}

\begin{proof}
  There is a 1-morphism $\Kbmo{X}{\tau}\rightarrow \Kbm{X}{\tau}$
  where $\Kbm{X}{\tau}$ is the refined functor defined in ~\cite{BM}.
  In ~\cite{BM} it is proved that $\Kbm{X}{\tau}$ is a proper
  Deligne-Mumford stack over $\CC$.  And it is clear that
  $\Kbmo{X}{\tau}\rightarrow \Kbm{X}{\tau}$ is a representable
  morphism which is an open immersion.  Thus $\Kbmo{X}{\tau}$ is a
  Deligne-Mumford stack which is separated and finite type over $\CC$.
\end{proof}

\subsection{Properties and Related Constructions}~\label{subsec-props}

Notice that with our notation $\Kbm{X}{\tau_r(e)}$ is the moduli stack
of Kontsevich stable maps $\Kgnb{0}{r}{X}{e}$, and
$\Kbmo{X}{\tau_r(e)}$ simply parametrizes those stable maps such that
the domain curve is irreducible.

\begin{defn}~\label{defn-evmor}
  Suppose that $\tau$ is a stable $A$-graph and
  $f\in\text{Flag}(\tau)$.  Then there is a $1$-morphism
\begin{equation}
\text{ev}_f:\Kbm{X}{\tau}\rightarrow X
\end{equation}
defined by sending a family of $\tau$-maps, $\eta$ (with notation as
above), to the morphism $h_{\partial f}\circ q_f$.
\end{defn}

If $\alpha=(\alpha_F,\alpha_V):\sigma\rightarrow\tau$ is a
combinatorial morphism of 
graphs, $\tau\hookleftarrow \sigma$, there is an associated
$1$-morphism
\begin{equation}
        \Kbm{X}{\alpha}:\Kbm{X}{\tau}\rightarrow\Kbm{X}{\sigma}.
\end{equation}
If $\alpha$ is the inclusion of $\sigma$ as a subgraph of $\tau$, then
$\Kbm{X}{\alpha}$ is the \emph{forgetful morphism} which ``remembers'' only
those components of $\tau$-maps whose vertex is contained in $\sigma$.
The reader is referred to~\cite[Theorem 3.6]{BM} for the precise
definition.  We will refer to the restriction of $\Kbm{X}{\alpha}$ to
$\Kbmo{X}{\tau}$ as $\Kbmo{X}{\alpha}$.

If $\phi=(\phi_W,\phi^F):\tau\rightarrow \tau'$ is a contraction of
stable $A$-graphs, there is a corresponding 1-morphism of proper
Deligne-Mumford stacks
\begin{equation}
\Kbm{X}{\phi}:\Kbm{X}{\tau}\rightarrow \Kbm{X}{\tau'}.
\end{equation}
This morphism ``forgets'' the labeling of some the individual
components of the domain curve.  The reader is referred
to~\cite[Theorem 3.6]{BM} for the precise definition.  We will denote
by $\Kbmo{X}{\phi}$ the restriction of this 1-morphism to the open
substack $\Kbmo{X}{\tau}$ of $\Kbm{X}{\tau}$.

One important case to understand is when $\beta(\tau)=0$.  We have
already defined $\Kbmo{X}{\tau_\emptyset}=\Kbm{X}{\tau_\emptyset}=X$
where $\tau_\emptyset$ is the empty graph.  For any stable $A$-graph
$\tau$ such that $\beta(\tau)=0$ and such that
$\#\text{Tail}(\tau)=r$, we have $\Kbmo{X}{\tau}= X \times
\Kbmo{*}{\tau}$ where $\Kbmo{*}{\tau}\subset \Mgnb{0}{r}$ is the
obvious substack.

Consider the case that $\phi:\tau\rightarrow \tau'$ is a contraction
of stable A-graphs such that $\beta(\tau)=\beta(\tau')=0$.  Then
$\Kbmo{X}{\tau}\rightarrow \Kbm{X}{\tau'}$ is simply the product of
$\text{id}_X:X\rightarrow X$ with the 1-morphism
$\Kbmo{*}{\phi}:\Kbmo{*}{\tau}\rightarrow \Kbm{*}{\tau'}$.  In
particular, using the notation of ~\cite{BM}, consider the case that
$\phi$ is an \emph{isogeny}, i.e. $\phi$ is the morphism which removes
some subset of the set of tails from $\tau$ and then stabilizes the
resulting (possibly unstable) graph.

\begin{lem}~\label{lem-isog}  Let $\tau,\tau'$ be stable $A$-graphs such
  that $\beta(\tau)=\beta(\tau')=0$ and let $\phi:\tau\rightarrow
  \tau'$ be an isogeny.  Then
  $\Kbmo{X}{\phi}:\Kbmo{X}{\tau}\rightarrow \Kbm{X}{\tau'}$ is smooth
  of relative dimension $\text{dim}(X,\tau)-\text{dim}(X,\tau')$ with
  geometrically connected fibers.
\end{lem}

\begin{proof}  Of course it is equivalent to prove that
\begin{equation}
\Kbmo{*}{\phi}:\Kbmo{*}{\tau}\rightarrow \Kbm{*}{\tau'}
\end{equation}
is smooth of relative 
dimension $\text{dim}(X,\tau)-\text{dim}(X,\tau')$ with geometrically
connected fibers.  Now it follows by Proposition 7.4 of
~\cite{BM} that $\Kbmo{*}{\tau}$ and $\Kbm{*}{\tau'}$ have the
expected dimension.  Thus all we really need to show is that
$\Kbmo{*}{\tau}\rightarrow \Kbm{*}{\tau'}$ is smooth with geometrically
irreducible fibers.  Moreover, since every isogeny is a composition of
morphisms obtained by stably removing one tail, we may suppose that
$\phi:\tau\rightarrow \tau'$ corresponds to stably removing one tail
$f\in\text{Tail}(\tau)$.  

There are two cases.  Suppose first of all that when we remove $f$
from $\tau$, the resulting graph is unstable.  But then $\partial f =
v$ is a vertex with valence 3.  Since a rational curve with 3 marked
points has no moduli, we conclude that $\Kbmo{*}{\tau}\rightarrow
\Kbm{*}{\tau'}$ is an open immersion.

The second case is that when we remove $f$ from $\tau$, the resulting
graph is stable, i.e. the resulting graph is just $\tau'$.  But then
if $v=\phi(\partial f)$, we conclude that
$\phi:\Kbmo{*}{\tau}\rightarrow \Kbm{*}{\tau'}$ is simply an open
subset of the universal curve over $\Kbm{*}{\tau'}$ corresponding to
the vertex $v$.  In both cases we conclude that
$\Kbm{*}{\tau}\rightarrow \Kbm{*}{\tau'}$ is smooth with geometrically
connected fibers.
\end{proof}

For each stable $A$-graph $\tau$ define $e=\beta(\tau)$ and define
$r=\#\text{Tail}(\tau)$.  Then there is a contraction
$\phi:\tau\rightarrow \tau_r(e)$ which is unique up to a labeling of
the tails of $\tau$.

\begin{defn}~\label{defn-cancon}
  The contraction $\phi$ above is the \emph{canonical contraction}.
  The corresponding $1$-morphism (well-defined up to relabeling the
  tails)
\begin{equation}
\Kbmo{X}{\phi}:\Kbmo{X}{\tau}\rightarrow\Kgnb{0}{r}{X}{e}
\end{equation}
will be referred to as the \emph{canonical contraction morphism}.
\end{defn}

Notice that the image of $\Kbmo{X}{\phi}$ as a subset of the set
$|\Kgnb{0}{r}{X}{e}|$ is well-defined.

\begin{prop}~\label{prop-locclo}
  Let $\phi:\tau\rightarrow \tau'$ be a contraction of stable
  $A$-graphs.  The image of the 1-morphism $\Kbmo{X}{\phi}$ is a
  locally closed subset of the topological space $|\Kbm{X}{\tau}|$.
\end{prop}

\begin{proof}  For notation's sake let's denote the continuous map
  of topological spaces
\begin{equation}
|\Kbm{X}{\phi}|:|\Kbm{X}{\tau}|\rightarrow
|\Kbm{X}{\tau'}|
\end{equation}
by $ f: M\rightarrow M'$ and let's denote the
open substack $\Kbmo{X}{\tau}$ of $\Kbm{X}{\tau}$ by $M^o$.  Then
$f:M\rightarrow M'$ is a closed map.  And it is easy to see that
$f^{-1}(f(M^o)) = M^o$.  Therefore $f(M^o)= f(M) - f(M - M^o)$ is
a difference of closed sets and so is locally closed.
\end{proof}

We now fix $n$ and $\alpha$ and consider the set $S$ of all images
\begin{equation}
\lt\{\text{Image}(\Kbmo{X}{\phi})\rt\}
\end{equation}
as $\phi$ ranges
over all contractions of stable $A$-graphs to $\tau_n(\alpha)$.
The set of isomorphism classes of such contractions is clearly
finite.  The previous lemma shows that $S$ forms a \emph{locally
closed decomposition} of the topological space
$|\Kgnb{0}{n}{X}{\alpha}|$, i.e. a partition of
$|\Kgnb{0}{n}{X}{\alpha}|$ into locally closed subsets.  This
partition is what we call the \emph{Behrend-Manin decomposition}.

\section{Flatness and Dimension Results}~\label{sec-flat}

In this section we consider the dimensions of the stacks
$\Kbmo{X}{\tau}$ and the evaluation morphisms.  The main property we
are interested in is the following:

\begin{defn}~\label{defn-prop1}
  Given a stable $A$-graph $\tau$, we say that $\cD(X,\tau)$ holds if
  the dimension of every irreducible component of $\Kbmo{X}{\tau}$
  equals the expected dimension $\text{dim}(X,\tau)$.
\end{defn}

By deformation theory there is an a priori lower bound on the
dimension of any irreducible component of $\Kbmo{X}{\tau}$:

\begin{lem}~\label{lem-lbdim}
  Every irreducible component of $\Kbm{X}{\tau}$ has dimension at
  least $\text{dim}(X,\tau)$.  In particular, every irreducible
  component of $\Kbmo{X}{\tau}$ has dimension at least
  $\text{dim}(X,\tau)$.
\end{lem}

\begin{proof}
  This is a standard result. In the case that $\tau=\tau_r(e)$ it
  follows from ~\cite[section 5.2]{FP}.  In the general case the
  theorem follows from ~\cite{B}.  The theorem isn't actually stated
  in ~\cite{B}, so we show how it follows from results there.

Let $\mathfrak{M}(\tau)$ denote the stack of $\tau$-marked prestable
curves as in ~\cite{B}.  There is a forgetful $1$-morphism of
algebraic (Artin) stacks
\begin{equation}
        \Kbm{X}{\tau}\rightarrow \mathfrak{M}(\tau).
\end{equation}
By ~\cite[Lemma 1]{B}, $\mathfrak{M}(\tau)$ is smooth of dimension
\begin{equation}
        \#\text{Tail}(\tau) - \#\text{Edge}(\tau) - 3.
\end{equation}
Let $\mathfrak{C}(\tau)\rightarrow \mathfrak{M}(\tau)$ be the
universal curve.  By ~\cite[Proposition 4]{B}, $\Kbm{X}{\tau}$ is an
open substack of the relative morphism-scheme,
\begin{equation}
\text{Mor}_{\mathfrak{M}(\tau)}\lt(\mathfrak{C}(\tau),X\times
\mathfrak{M}(\tau)\rt).
\end{equation}
By ~\cite[Theorem 2.17.1]{K}, it follows that every irreducible
component of $\Kbm{X}{\tau}$ has dimension at least
\begin{equation}
        h^0(C,f^*T_X)-h^1(C,f^*T_X)
        +\#\text{Tail}(\tau)-\#\text{Edge}(\tau). 
\end{equation}
By Riemann-Roch, we have $(h^0-h^1)(C,f^*T_X) = -K_X.f_*[C] +
\text{dim}(X)$.  We conclude that every irreducible component of
$\Kbm{X}{\tau}$ has dimension at least
\begin{equation}
        -K_X.f_*[C] + \text{dim}(X) - 3 + \#\text{Flag}(\tau) -
         \#\text{Edge}(\tau) = \text{dim}(X,\tau).
\end{equation}
\end{proof}

When $\text{Vertex}(\tau)$ has more than one element, we can try to
reduce $\cD(X,\tau)$ to $\cD(X,\tau_i)$ for some proper subgraphs
$\tau_i$ of $\tau$, thus giving an inductive proof that $\cD(X,\tau)$
holds.  To carry out such a proof, we need to know that the evaluation
morphisms have constant fiber dimension.  So we introduce the
following property:

\begin{defn}~\label{defn-prop2}
  Given a stable $A$-graph $\tau$ and a flag $f\in\text{Flag}(\tau)$,
  we say that $\cE(X,\tau,f)$ holds if
\begin{equation}
\text{ev}_f:\Kbmo{X}{\tau}\rightarrow X
\end{equation}
is dominant and has constant fiber dimension
$\text{dim}(X,\tau)-\text{dim}(X)$.  
\end{defn}

Notice that if there is any flag $f\in\text{Flag}(\tau)$ such that
$\cE(X,\tau,f)$ holds, then it follows that $\cD(X,\tau)$ holds.

In the case that $X\subset \PP^N$ is a complete intersection, then the
properties $\cD$ and $\cE$ are equivalent to stronger properties.

\begin{defn}~\label{defn-prop1'}
  Given a stable $A$-graph $\tau$, we say that $\lci(X,\tau)$ holds if
  $\Kbmo{X}{\tau}$ is a local complete intersection and if
  $\cD(X,\tau)$ holds.  Given a stable $A$-graph $\tau$ and a flag
  $f\in\text{Flag}(\tau)$, we say that $\fe(X,\tau,f)$ holds if
\begin{equation}
\text{ev}_f:\Kbmo{X}{\tau} \rightarrow X
\end{equation}
is flat of relative dimension $\text{dim}(X,\tau)-\text{dim}(X)$.
\end{defn}

\begin{lem}~\label{lem-propeq}
  If $X\subset \PP^N$ is a complete intersection, then $\cD(X,\tau)$
  holds iff $\lci(X,\tau)$ holds.  Also $\cE(X,\tau,f)$ holds iff
  $\fe(X,\tau,f)$ holds.  The same result holds with $\Kbmo{X}{\tau}$
  replaced by $\Kbm{X}{\tau}$.
\end{lem}

\begin{proof}  
  Suppose that $X$ is a complete intersection of $r = N-n$
  hypersurfaces of degrees $d_1,\dots,d_r$.  Consider
  $\Kbmo{\PP^N}{\tau}$ and denote the universal curve by
\begin{equation}
\pi:\mathcal{C}\rightarrow \Kbmo{\PP^N}{\tau}.
\end{equation}
Let $h:\mathcal{C}\rightarrow \PP^N$ denote the universal map.  Since
\begin{equation}
\OO_{\PP^N}({\mathbf d}) := \OO_{\PP^N}(d_1)\oplus \dots \oplus
\OO_{\PP^N}(d_r) 
\end{equation}
is generated by global sections, also $h^*\OO_{\PP^N}({\mathbf d})$ is
generated by global sections.  On a genus 0 tree, if $F$ is a sheaf
generated by global sections then $H^1(C,F)=0$, so
$R^1\pi_*\lt(h^*\OO_{\PP^N}({\mathbf d})\rt) = 0$.

Now by ~\cite[Proposition 7.4]{BM}, $\Kbmo{\PP^N}{\tau}$ is smooth of
dimension
\begin{equation}
(N+1)\beta(\tau) + (N-3) +
\#\text{Flag}(\tau) - \#\text{Edge}(\tau).
\end{equation}
So by ~\cite[Corollary II.2]{AVar} the pushforward
$E:=\pi_*\lt(h^*\OO_{\PP^N}({\mathbf d})\rt)$ is a locally free sheaf
of rank
\begin{equation}
 \sum_{i=1}^r \chi(C,h^*\OO_{\PP^N}(d_i)) =
\sum_{i=1}^r \lt(d_i\beta(\tau) + 1\rt) = \lt(\sum_i d_i\rt)\beta(\tau) + r.
\end{equation}
Now the defining equations of the hypersurfaces in $\PP^N$ give a
global section $\sigma$ of $E$, and $\Kbmo{X}{\tau}$ is precisely the
zero scheme of $\sigma$.  Finally notice that the expected
codimension, $\text{dim}(\PP^n,\tau) - \text{dim}(X,\tau)$, of
$\Kbmo{X}{\tau}$ in $\Kbmo{\PP^N}{\tau}$ is just
\begin{equation}
-K_{\PP^N}.f_*[C] +
K_X.f_*[C] + \text{dim}(\PP^N) - \text{dim}X = \lt(\sum_i
d_i\rt)\beta(\tau) + r.
\end{equation}
Thus, if $\Kbmo{X}{\tau}$, then it follows that $\Kbmo{X}{\tau}$ is a
local complete intersection.  So if $\cD(X,\tau)$ holds, then also
$\lci(X,\tau)$ holds.  The opposite inclusion is obvious.

Now suppose that $\cE(X,\tau,f)$ holds.  In particular $\cD(X,\tau)$
holds, so $\lci(X,\tau)$ holds.  But now by ~\cite[Theorem 23.1]{Ma},
$\text{ev}_f$ is a dominant morphism from a Cohen-Macaulay scheme to a
smooth scheme with constant fiber dimension, therefore it is flat.  So
$\fe(X,\tau,f)$ holds.  The opposite inclusion is obvious.

The same proof works when we replace $\Kbmo{X}{\tau}$ by $\Kbm{X}{\tau}$.
\end{proof}


Consider the following diagram:

\begin{picture}(250,200)
\put(50,50){\makebox(200,100){\includegraphics{Graph1b.eps}}}
\put(150,20){\makebox(0,0)[b]{Diagram 1}}
\end{picture}

Here $\tau$ is an $A$-graph which contains the two subgraphs 
\begin{equation}
a_1:\tau\hookleftarrow \tau_1,\ \  a_2:\tau\hookleftarrow \tau_2.
\end{equation}
The edge $\{f_1,f_2\}$ of $\tau$ is made up of the two tails
$f_1\in\text{Tail}(\tau_1), f_2\in\text{Tail}(\tau_2)$.  Let
$f\in\text{Flag}(\tau_2)$ be any flag (possibly $f=f_2$).

\begin{lem}~\label{lem-red1}
If $\fe(X,\tau_1,f_1)$ and $\fe(X,\tau_2,f)$
hold, then $\fe(X,\tau,f)$ holds.
\end{lem}

\begin{proof}
The combinatorial morphisms $a_1$ and $a_2$ give rise to a $1$-morphism 
\begin{equation}
\Kbmo{X}{a_1,a_2}:\Kbmo{X}{\tau}\rightarrow
\Kbmo{X}{\tau_1}\times_{\text{ev}_{f_1}, X, \text{ev}_{f_2}}
\Kbmo{X}{\tau_2}.
\end{equation}
It is clear from the definition of strict $\tau$-maps that
$\Kbmo{X}{a_1,a_2}$ is an open immersion.  

Since $\text{ev}_{f_1}:\Kbmo{X}{\tau_1}\rightarrow X$ is flat of
relative dimension 
$\text{dim}(X,\tau_1) - \text{dim}(X)$, it follows that the projection
morphism 
\begin{equation}
\text{pr}_2: \Kbmo{X}{\tau_1}\times_{\text{ev}_{f_1},X, \text{ev}_{f_2}}
\Kbmo{X}{\tau_2} \rightarrow \Kbmo{X}{\tau_2}
\end{equation}
is flat of relative dimension $\text{dim}(X,\tau_1)-\text{dim}(X)$.
And $\text{ev}_{f}:\Kbmo{X}{\tau_2}\rightarrow X$ is flat of relative
dimension 
$\text{dim}(X,\tau_2) - \text{dim}(X)$.  Thus the composite morphism
\begin{equation}
\Kbmo{X}{\tau_1}\times_{\text{ev}_{f_1},X, \text{ev}_{f_2}}
\Kbmo{X}{\tau_2} \xrightarrow{\text{pr}_2} \Kbmo{X}{\tau_2}
\xrightarrow{\text{ev}_{f}} X
\end{equation}
is flat of relative dimension $\text{dim}(X,\tau_1) +
\text{dim}(X,\tau_2) - 2\text{dim}(X)$.
Of course $\text{ev}_f:\Kbmo{X}{\tau}\rightarrow X$ is simply the
restriction of the composite morphism, so it is flat of the same
relative dimension.
But notice that
\begin{equation}
\begin{array}{c}
 \text{dim}(X,\tau_1)+\text{dim}(X,\tau_2) - \text{dim}(X) = \\
 \lt(-K_X.\beta(\tau_1) + -K_X.\beta(\tau_2)\rt) +
2\lt(\text{dim}(X)-3\rt) + \\
\lt(\#\text{Flag}(\tau_1) +
\#\text{Flag}(\tau_2)\rt) + \lt(\#\text{Edge}(\tau_1) +
\#\text{Edge}(\tau_2)\rt) - \text{dim}(X)= \\
-K_X.\beta(\tau) + 2\lt(\text{dim}(X) - 3\rt) +
\lt(\#\text{Tail}(\tau) + 2\rt) - \\
\lt(\#\text{Edge}(\tau) - 1\rt)
-\text{dim}(X) = \text{dim}(X,\tau).
\end{array}
\end{equation}
From this it follows that  
\begin{equation}
\text{ev}_{f}:\Kbmo{X}{\tau}\rightarrow X
\end{equation}
is flat of constant fiber dimension
$\text{dim}(X,\tau) - \text{dim}(X)$.  Thus
$\fe(X,\tau,f)$ holds.  
\end{proof}

\begin{defn}~\label{defn-E}
Suppose that $\tau$ is a stable $A$-graph and define the
\emph{maximum component degree} of $\tau$ to be
\begin{equation}
E(\tau) = \sup_{v\in\text{Vertex}(\tau)} \beta(v).
\end{equation}
\end{defn}


\begin{prop}~\label{prop-red}
Suppose that $\tau$ is a stable $A$-graph with $E(\tau)=E$.  
If for each $e=0,\dots,E$ we have $\fe(X,\tau_1(e),f)$
holds, then for each flag $f\in\text{Flag}(\tau)$,
$\fe(X,\tau,f)$ holds.
\end{prop}

\begin{proof}
We prove this by induction on $\#\text{Vertex}(\tau)$.  Suppose $\tau$
has a single vertex.  Then $\tau=\tau_r(e)$ for some $r$ and $e$.  If
$e=0$, then $\Kbm{X}{\tau_r(0)}=\Kbm{*}{\tau_r(0)}\times X$, and the
evaluation morphism is just projection.  So $\text{ev}_f$ is flat of
relative dimension 
\begin{equation}
\text{dim}(\Kbm{*}{\tau_r(0)}) = \text{dim}(X,\tau_r(0)) -
\text{dim}(X),
\end{equation}
i.e.  $\fe(X,\tau_r(0),f)$ holds.  

Next consider the case $\tau=\tau_r(e)$ with 
$e>0$.  For any flag $f\in \text{Flag}(\tau_r(e))$ there is a
combinatorial morphism 
$a:\tau_{r}(e)\hookleftarrow \tau_1(e)$ which maps the unique flag 
$f_1\in\text{Flag}(\tau_1(e))$ to $f$.  The associated $1$-morphism
\begin{equation}
\Kbmo{X}{a}:\Kbmo{X}{\tau_r(e)}\rightarrow \Kbmo{X}{\tau_1(e)}
\end{equation}
is isomorphic to an open subset of the $(e-1)$-fold fiber product of
the universal curve over $\Kbmo{X}{\tau_1(e)}$.  Since the universal
curve is smooth of relative dimension $1$ over $\Kbmo{X}{\tau_1(e)}$,
we conclude that $\Kbmo{X}{a}$ is smooth of relative dimension $e-1$.  
The evaluation morphism
$\text{ev}_f:\Kbmo{X}{\tau_r(e)}\rightarrow X$ factors as the
composition
\begin{equation}
\begin{CD}
\Kbmo{X}{\tau_r(e)} @>\Kbmo{X}{a} >> \Kbmo{X}{\tau_1(e)} @>\text{ev}_{f_1}
>> X.
\end{CD}
\end{equation}
Therefore if $\fe(X,\tau_1(e),f_1)$ holds, then
$\text{ev}_f$ is flat of relative dimension
\begin{equation}
(e-1) + \lt(\text{dim}(X,\tau_1(e)) - \text{dim}(X)\rt) =
\text{dim}(X,\tau_r(e)) - \text{dim}(X),
\end{equation}
in other words $\fe(X,\tau_r(e),f)$ holds.
Thus the Proposition is proved when $\tau$ has a single vertex.

Now suppose that $\tau$ has more than one vertex and suppose that for
all $e\leq E=E(\tau)$, $\fe(\tau_1(e),f)$ holds.  By
way of induction, assume that the
proposition is true for all graphs $\tau'$ such that
$\#\text{Vertex}(\tau') < \#\text{Vertex}(\tau)$.  
Let $\{f_1,f_2\}$
be any edge.  Define $\tau_1$ and $\tau_2$ to be the graphs obtained
by breaking the edge into two tails (see diagram 1).  
Let $f\in\text{Flag}(\tau)$ be a
flag, and without loss of generality suppose that
$f\in\text{Flag}(\tau_2)$.  Now $E(\tau_2) \leq E(\tau)$ and
$\#\text{Vertex}(\tau_2) < \#\text{Vertex}(\tau)$, so by the
induction assumption $\fe(X,\tau_2,f)$ holds.  Also
$E(\tau_1)\leq E(\tau)$ and $\#\text{Vertex}(\tau_1) <
\#\text{Vertex}(\tau)$, so by the induction assumption
$\fe(X,\tau_1,f_1)$ holds.  Then by Lemma~\ref{lem-red1},
we conclude that $\fe(X,\tau,f)$ holds.  So the proposition
is proved by induction.
\end{proof}

\section{Specializations}

In the previous section we reduced the flatness and dimension results for
a general stable $A$-graph $\tau$ to flatness and dimension results
for the the stable $A$-graphs $\tau_1(e)$ with $0\leq e\leq
E(\tau).$
In this
section we will use specializations to reduce the flatness and
dimension results for all $\tau_1(e), e> 1$ to flatness and dimension
results for a finite number of cases $\tau_1(e), e=1,\dots,E(X)$ where
$E(X)$ is the \emph{threshold degree} of $X$.  We define a stable
$A$-graph $\sigma$ to be \emph{basic} if for each vertex
$v\in\text{Vertex}(\sigma)$, we have
$\beta(v)\leq E(X)$.  The
specializations we produce will show that every irreducible component
of $\Kbm{X}{\tau}$ contains a \emph{basic} locally closed subset
$\Kbmo{X}{\sigma}$.  Thus to understand the
irreducible components of $\Kbm{X}{\tau}$ it suffices to understand
the irreducible components which pass through the general point of a
basic locus $\Kbmo{X}{\sigma}$.

\textbf{Convention}  Suppose that we have a contraction
$\phi:\sigma\rightarrow \tau$.  There is an induced morphism
\begin{equation}
\Kbmo{X}{\phi}:\Kbmo{X}{\sigma}\rightarrow \Kbm{X}{\tau}
\end{equation}
which is unramified with locally closed image.  We will speak of
$\Kbmo{X}{\sigma}$ as though it is a substack of $\Kbm{X}{\tau}$.
Thus given an irreducible component $M\subset \Kbmo{X}{\tau}$ and an
irreducible component $N\subset \Kbmo{X}{\sigma}$ we will say
$N\subset \overline{M}$ to mean that the image of $N$ is contained in
$\overline{M}$.  

The basic lemma is the following easy version of Mori's bend-and-break
lemma.  

\begin{lem}~\label{lem-bb}
Let $e>0$.
There is no complete curve contained in a fiber of the evaluation
morphism 
\begin{equation}
\text{ev}_{f_1,f_2}:\Kbmo{X}{\tau_2(e)}\rightarrow X\times X.
\end{equation}
\end{lem}

\begin{proof}
Suppose that $C$ is a complete curve and $\zeta:C\rightarrow
\Kbmo{X}{\tau_2(e)}$ has image in 
a fiber of $\text{ev}_{f_1,f_2}$.  Denote the family $\zeta$ of strict
$\tau_2(e)$-maps by $(\pi:\Sigma\rightarrow C,h,(q_1,q_2))$.
Denote
$\text{ev}_{f_1,f_2}(C)=(p_1,p_2)$.  Let $H\subset X$ be a hyperplane
section containing neither $p_1$ nor $p_2$.  Define $C'=\Sigma\times_X
H$, so $C'$ is a finite ramified cover of $C$.  Let $B$ be the
normalization of an irreducible component of $C'$ which dominates $C$.  
The base-change $\Sigma \times_C B$ now admits the two sections
$q_1,q_2$ as well as a third section $q_3$ which is everywhere
disjoint from both $q_1$ and $q_2$.  Any $\PP^1$-bundle with three
everywhere disjoint sections is isomorphic to $\PP^1\times B$ and the
three sections are constant sections $\{0\}\times B, \{1\}\times B,
\{\infty\}\times B$.  But now the morphism $h:\Sigma\times_C
B\rightarrow X$ contracts the sections $q_1$ and $q_2$.  By the
rigidity lemma, \cite[p. 43]{AVar}, we conclude that $h$ factors through
the projection $\Sigma\times_C B\cong \PP^1\times B \rightarrow
\PP^1$.  So we conclude that $C\rightarrow \Kbmo{X}{\tau}$ is
a constant map.
\end{proof}

\begin{cor}~\label{cor-bb}
Let $M\subset \Kbm{X}{\tau_2(e)}$ be an irreducible, closed substack and
suppose that the fibers of $\text{ev}_{f_1,f_2}:M\rightarrow X\times
X$ have dimension at least $1$.  Then $M\cap
\lt(\Kbm{X}{\tau_2(e)}- \Kbmo{X}{\tau_2(e)}\rt)$ is either all of $M$
or contains an irreducible component with codimension $1$ in $M$.
\end{cor}

\begin{proof}
Suppose that $M$ intersects $\Kbmo{X}{\tau_2(e)}$.  Define
$I\subset X\times X$ to be the image $I=\text{ev}_{f_1,f_2}(M)$.  In
order to prove that 
\begin{equation}
\partial M :=
M\cap\lt(\Kbm{X}{\tau_2(e)}-\Kbmo{X}{\tau_2(e)}\rt)
\end{equation}
has an irreducible component of codimension $1$ in $M$, it suffices to
prove that for every $(p_1,p_2)\in I$, $\text{ev}^{-1}(p_1,p_2)\cap
\partial M\subset \text{ev}^{-1}(p_1,p_2)\cap M$ has codimension $1$.  
Suppose that it has codimension at least $2$.  Then the coarse moduli
space $|\text{ev}^{-1}(p_1,p_2)\cap \partial M|\subset
|\text{ev}^{-1}(p_1,p_2) \cap M|$ has codimension at least $2$.  Since
the coarse moduli spaces are proper varieties, we can find a complete
curve $C$ in $|\text{ev}^{-1}(p_1,p_2)\cap M|$ which does not
intersect $|\text{ev}^{-1}(p_1,p_2)\cap \partial M|$.  Since
$\Kbm{X}{\tau_2(e)}$ is a Deligne-Mumford stack, there exists a finite
ramified cover $C'\rightarrow C$ such that $C'\rightarrow
|\Kbm{X}{\tau_2(e)}|$ factors through $\Kbm{X}{\tau_2(e)}\rightarrow
|\Kbm{X}{\tau_2(e)}|$.  But then by Lemma~\ref{lem-bb}, we conclude
that $C'\rightarrow |\Kbm{X}{\tau_2(e)}|$ is constant, which
contradicts the construction of $C$.  Therefore $\partial M\subset M$
has codimension $1$.
\end{proof}

The main application is the following:

\begin{prop}~\label{prop-bbflat}
Suppose that $X\subset \PP^N$ is a complete intersection.  
Suppose that $\fe(X,\tau_1(e),f_1)$ holds for every $e <
E$
and suppose that every irreducible component of $\Kbmo{X}{\tau_1(E)}$
has dimension at least $2\text{dim}(X)$.  Then
$\fe(X,\tau_1(E),f_1)$ holds as well.  
\end{prop}

\begin{proof}
By Lemma~\ref{lem-propeq}, to prove that
$\fe(X,\tau_1(E),f_1)$ holds, it suffices to prove that
$\cE(X,\tau_1(E),f_1)$ holds.  Let $\zeta\in
\Kbmo{X}{\tau_1(e)}$ be a point, denote $p=\text{ev}_{f_1}(\zeta)$,
and let $M\subset \text{ev}_{f_1}^{-1}(p)$ be an irreducible component
which contains $p$.  We need to prove that
$\text{dim}(M)=\text{dim}(X,\tau_1(e)) -\text{dim}(X)$.  

Now consider the forgetful morphism
\begin{equation}
\Kbmo{X}{a}:\Kbmo{X}{\tau_2(e)}\rightarrow \Kbmo{X}{\tau_1(e)}.
\end{equation}  
This is a smooth surjective morphism: let $N\subset
\Kbmo{X}{\tau_2(e)}$ be the preimage of $M$.  Then
$\text{dim}(N)=\text{dim}(M)+1$, thus we have to prove that 
\begin{equation}
\text{dim}(N) = \text{dim}(X,\tau_1(e))+1 - \text{dim}(X).
\end{equation}
It suffices to prove that the general fiber of $\text{ev}_{f_2}:N\rightarrow
X$ has dimension at most $\text{dim}(X,\tau_1(e))+1 - 2\text{dim}(X)$
(since we already know the dimension is at least this large).

Choose any point $q\neq p$ in $\text{ev}_{f_2}(N)$ and consider $N_q
:= \text{ev}_{f_2}^{-1}(q)\cap N$.  
By assumption, $\text{dim}(X,\tau_1(e))+1-2\text{dim}(X) \geq 1$, so
$\text{dim}(N_q) \geq 1$.  
Define $\overline{N}\subset \Kbm{X}{\tau_2(e)}$ to be the closure of
$N$.
By Corollary~\ref{cor-bb} we conclude that $\partial
\overline{N}\subset \overline{N}$ has codimension $1$.  In other
words, there is a stable $A$-graph $\sigma'\not\cong\tau_2(e)$ whose
canonical 
contraction is $\phi':\sigma'\rightarrow \tau_2(e)$ and such that
$\overline{N}\cap \text{Image}(\Kbmo{X}{\sigma'})\subset \overline{N}$
has an irreducible component of codimension $1$.  

Now there is precisely one stable $A$-graph $\sigma_0\not\cong
\tau_2(e)$ whose stabilization after removing $f_2$ equals
$\tau_1(e)$, namely:

\begin{picture}(250,150)
\put(50,50){\makebox(200,100){\includegraphics{Graph2.eps}}}
\put(150,20){\makebox(0,0)[b]{Diagram 2}}
\end{picture}

%include a diagram of ((d,\emptyset),(0,\{f_1,f_2\}))

By the assumption that $q\neq p$, the point $(q,p)\not\in
\text{ev}_{f_1,f_2}(\Kbmo{X}{\sigma_0})$.  So $\sigma'$ is
not $\sigma_0$.  We conclude that the image of
$\Kbmo{X}{\sigma'}$ in $\Kbm{X}{\tau_1(e)}$ (under the map which
stably removes $f_2$) is again a boundary component $\Kbm{X}{\sigma}$
for some $\phi:\sigma\rightarrow \tau_1(e)$ (not the identity).  Therefore
$\overline{M}\cap\text{Image}(\Kbmo{X}{\sigma})\subset\overline{M}$
is a locally closed substack such that some irreducible component has
codimension one in $\overline{M}$.  

Since $\sigma\not\cong\tau_1(E)$, we have $E(\sigma)< E$.  By our
assumption and by Proposition~\ref{prop-red}, we conclude that
$\fe(X,\sigma,f_1)$ holds.  In particular,
$\overline{M}\cap\text{Image}(\Kbmo{X}{\sigma})$ has dimension at most
$\text{dim}(X,\sigma)-\text{dim}(X)$.  So the
dimension of $\overline{M}$ is at most
$\text{dim}(X,\sigma)+1-\text{dim}(X)$.  Since
$\text{dim}(X,\sigma)+1\leq \text{dim}(X,\tau_1(e))$, we conclude
that $\text{dim}(M)\leq \text{dim}(X,\tau_1(e))-\text{dim}(X)$.  So
$\fe(X,\tau_1(e),f_1)$ holds.  
\end{proof}

\textbf{Remark}  The condition that $X\subset \PP^N$ be a complete
intersection is essentially superfluous.  If instead we had worked
throughout with the property $\cE(X,\tau,f)$ rather than
$\fe(X,\tau,f)$ (which is a little trickier), then the
argument above proves the analogous result without this condition on
$X$.  

\begin{defn}~\label{defn-thrdeg}
If $X\subset \PP^N$ is a complete intersection of hypersurfaces of
degrees $d_1,d_2,\dots,d_r$, define the \emph{threshold degree} to be
\begin{equation}
E(X)=E(N,(d_1,\dots,d_r)) = \lt\lfloor \frac{N+2-r}{N+1-(d_1+\dots+d_r)}
\rt\rfloor.
\end{equation}
In particular, if $X\subset \PP^N$ is a hypersurface of degree $d <
\frac{N+1}{2}$ then $E(X) = 1$.
\end{defn}

\begin{cor}~\label{cor-bbflat}
Suppose that $X\subset \PP^N$ is a complete intersection of
hypersurfaces of degrees $d_1,d_2,\dots,d_r$.  If
$\fe(X,\tau_1(e),f_1)$ holds for each $1\leq e \leq E(X)$,
then for every stable $A$-graph $\tau$ and flag
$f\in\text{Flag}(\tau)$, $\fe(X,\tau,f)$ holds.
In particular, if $X\subset \PP^N$ is a hypersurface of degree
$d<\frac{N+1}{2}$ then it suffices to prove
$\fe(X,\tau_1(1),f_1)$.  
\end{cor}

\begin{proof}
By Proposition~\ref{prop-red}, to prove that
$\fe(X,\tau,f)$ always holds it suffices to prove that
$\fe(X,\tau_1(e),f_1)$ always holds (for $e>0$).  Now for
$e>E(X)$ we have by Lemma~\ref{lem-lbdim} that every irreducible
component of $\Kbmo{X}{\tau_1(e)}$ has dimension at least
\begin{equation}
\begin{array}{c}
 \lt(N+1-(d_1+\dots +d_r)\rt)e + (N-r-3) + 1 \geq \\ 
\lt(N+1-(d_1+\dots+ d_r)\rt)\frac{N+2-r}{N+1-(d_1+\dots+d_r)} +
(N-r-3) + 1,
\end{array}
\end{equation}
which, of course, is just $2\text{dim}(X)$.  
So by Proposition~\ref{prop-bbflat} and induction, to
prove $\fe(X,\tau_1(e),f_1)$ for all $e$, it suffices to
prove $\fe(X,\tau_1(e),f_1)$ in the cases $e=1,\dots,E(X)$.
\end{proof}

\begin{cor}~\label{cor-bbflat2}
With the same hypotheses as in Corollary~\ref{cor-bbflat}, suppose
that $\fe(X,\tau_1(e),f_1)$ holds for each $1\leq e\leq
E(X)$.  Then for every stable $A$-graph $\tau$, $\Kbmo{X}{\tau}$ has
pure dimension $\text{dim}(X,\tau)$.  
\end{cor}

\begin{proof}
If $E(\tau)=0$, then $\Kbmo{X}{\tau}=\Kbmo{*}{\tau}\times X$ and the
result follows from ~\cite[Proposition 7.4]{BM}.  
Suppose that $E(\tau)>0$ and let
$v\in\text{Vertex}(\tau)$ be such that $\beta(v)>0$.  Define
$a:\tau'\hookleftarrow \tau$ to be the combinatorial morphism which
adds a new tail $f$ to $v$.  Then $\Kbmo{X}{a}$ is smooth and
surjective of relative
dimension $1$.  By Corollary~\ref{cor-bbflat},
$\fe(X,\tau',f)$ holds.  In particular, $\Kbmo{X}{\tau'}$
has pure dimension $\text{dim}(X,\tau')$.  It follows that
$\Kbmo{X}{\tau}$ has pure dimension $\text{dim}(X,\tau')-1 =
\text{dim}(X,\tau)$.  
\end{proof}

A second application of the proof of Proposition~\ref{prop-bbflat} is
the following:

\begin{prop}~\label{prop-bbspec}
Suppose that $X\subset \PP^N$ is a complete intersection.  Suppose
that $\fe(X,\tau_1(e),f_1)$ holds for every $e< E$ and
suppose that every irreducible component of $\Kbmo{X}{\tau_1(E)}$ has
dimension at least $2\text{dim}(X)$.  Then for every irreducible
component $M\subset\Kbmo{X}{\tau_0(E)}$ there is a graph
$\sigma=\tau_{0,0}(i,j)$, $0<i,j$ and $i+j=E$, and an irreducible
component $N\subset \Kbmo{X}{\tau_{0,0}(i,j)}$ such that $N\subset
\overline{M}$ is a codimension $1$ subvariety.
\end{prop}

\begin{proof} 
Let $M'\subset \Kbmo{X}{\tau_1(e)}$ be the irreducible component which
dominates $M$.  By the proof of Proposition~\ref{prop-bbflat} 
there is a graph $\sigma\not\cong\tau_1(e)$ with canonical
contraction $\phi:\sigma\rightarrow \tau_1(e)$ such that
$\overline{M'}\cap \Kbmo{X}{\sigma}\subset \overline{M}$ has
codimension $1$.  Now $\Kbmo{X}{\sigma}$ has dimension
$\text{dim}(X,\sigma)$ and by Lemma~\ref{lem-lbdim} $\overline{M'}$ has
dimension at least 
$\text{dim}(X,\tau_1(e))$.  But $\text{dim}(X,\sigma) =
\text{dim}(X,\tau_1(e)) - \#\text{Edge}(\sigma)$.  So 
$\sigma$ has exactly one edge, i.e. $\sigma=\tau_{1,0}(i,j)$ for some
$i,j$ with $i+j=E$.  By stability $i,j>0$.  Moreover,
$\overline{M'}\cap \Kbmo{X}{\sigma}$ has dimension
$\text{dim}(X,\sigma)$ so there is an irreducible
component $N'\subset \Kbmo{X}{\sigma}$ such that $N'\subset
\overline{M'}$.  

Since $\Kbmo{X}{\tau_{1,0}(i,j)}\rightarrow
\Kbmo{X}{\tau_{0,0}(i,j)}$ is smooth of relative dimension $1$, there
is an irreducible component $N\subset \Kbmo{X}{\tau_{0,0}(i,j)}$ such
that $N'$ is the preimage of $N$.  Thus $N\subset \overline{M}$ is a
codimension $1$ subvariety.
\end{proof}

\begin{defn}~\label{defn-basic}
Let $X\subset \PP^N$ be a complete intersection with
threshold degree
$E(X)=E$.  A stable $A$-graph $\tau$ is \emph{basic} for $X$ if its
maximal component degree $E(\tau)$ satisfies
$E(\tau)\leq E(X)$.  
\end{defn}

\begin{defn}~\label{defn-nice}
For an $A$-graph $\tau$, define its \emph{degree 0
subgraph} to be the maximal subgraph $\tau\hookleftarrow \tau^0$ such
that $E(\tau^0)=0$, i.e. $\tau^0$ is the (possibly disconnected)
subgraph of $\tau$ with 
\begin{equation}
\text{Vertex}(\tau^0)=\{v\in\text{Vertex}(\tau)| \beta(v)=0\}
\end{equation}
and 
\begin{equation}
\text{Flag}(\tau^0)=\{f\in\text{Flag}(\tau)| \partial
f\in\text{Vertex}(\tau^0)\}.
\end{equation}
A contraction of $A$-graphs $\phi:\sigma\rightarrow \tau$ is
\emph{nice} if $\phi$ induces an isomorphism $\sigma^0\cong\tau^0$.
\end{defn}

\begin{thm}~\label{thm-spec}
Let $X\subset \PP^N$ be a complete intersection, $\tau$ a stable
$A$-graph and $M\subset \Kbmo{X}{\tau}$ an irreducible component.
Suppose that $\fe(X,\tau_1(e),f_1)$ holds for each $1\leq
e\leq E(X)$.  Then there exists a nice contraction
$\phi:\sigma\rightarrow \tau$ and an irreducible component $N\subset
\Kbmo{X}{\sigma}$ such that $\sigma$ is basic and such that $N\subset
\overline{M}$.
\end{thm} 

\begin{proof}
We will prove this by induction on the maximal component degree
$E(\tau)$.  If $E(\tau) \leq E(X)$, then we can take $\phi$ to be the
identity $\tau\rightarrow \tau$ and $N=M$.  

Suppose that $E > E(X)$.  By way of induction, 
assume the theorem is proved for all
graphs $\tau$ with $E(\tau) < E$.  We will deduce the theorem for
graphs with $E(\tau)=E$ by induction on $\#\text{Vertex}(\tau)$, and
thus establish the theorem by induction on $E(\tau)$.

First we consider the case that $\#\text{Vertex}(\tau)=1$,
i.e. $\tau=\tau_r(E)$ 
for some $r$.  Define $a:\tau_r(E)\hookleftarrow \tau_0(E)$ be the
combinatorial morphism which strips the tails from $\tau_r(E)$.  
Then $\Kbmo{X}{a}:\Kbmo{X}{\tau_r(E)}\rightarrow
\Kbmo{X}{\tau_0(E)}$ is smooth, surjective with connected fibers of
dimension $r$.  So we 
conclude that $M$ is the preimage of an irreducible component
$M'\subset \Kbmo{X}{\tau_0(E)}$.  Now by Proposition~\ref{prop-bbspec}
there is a nice contraction $\psi:\rho\rightarrow \tau_0(E)$ and an
irreducible component $L\subset\Kbmo{X}{\rho}$ such that $L\subset
\overline{M}'$.  As $E(\rho)<E$, by assumption there exists
a nice contraction $\phi':\sigma'\rightarrow \rho$ such that $\sigma'$ is
basic and there exists an irreducible component $N'\subset
\Kbmo{X}{\sigma}$ such that $N'\subset \overline{L}$.  Therefore
$\psi\circ\phi':\sigma'\rightarrow \tau_0(E)$ is a nice contraction and
$N'\subset \overline{M'}$.  Now let $v\in\text{Vertex}(\sigma')$ be
any vertex and let $b:\sigma\hookleftarrow \sigma'$ be the graph
obtained by attaching $r$ tails to $v$.  Note that $\Kbmo{X}{b}$ is
smooth, surjective with connected fibers of dimension $r$.  
Let $\phi:\sigma\rightarrow
\tau_r(E)$ be the contraction obtained from $\psi\circ\phi'$ by
sending the $r$ tails of $\sigma$ to the $r$ tails of $\tau_r(E)$.
Then $\phi$ is a nice contraction and $\sigma$ is basic.  Define
$N\subset \Kbmo{X}{\sigma}$ to be the preimage of $N'$ under
$\Kbmo{X}{a}$.  
The morphism $\Kbm{X}{a}$ is compatible with $\Kbmo{X}{b}$,
i.e. $\Kbm{X}{a}\circ \Kbmo{X}{\phi} =
\Kbmo{X}{\psi\circ\phi'}\circ\Kbmo{X}{b}$, and $\Kbm{X}{a}$ is smooth
along the image of $\Kbmo{X}{\phi}$.  Thus we conclude that $N\subset
\overline{M}$, the theorem is proved for $M$.

Now we consider the general case.  
For each graph $\tau$, define
\begin{equation}
l(\tau):= \#\text{Vertex}(\tau).
\end{equation}
When $l=1$, we have the case in the last paragraph.  Suppose $l>1$
and, by way of induction on $l$, 
suppose that for all stable $A$-graphs $\tau$ with
$E(\tau)=E$ and with $l(\tau) < l$, the theorem is proved.
Suppose that $\tau$ is a stable $A$-graph with $E(\tau)=E$
and $l(\tau)=l$.  Let $\{f_1,f_2\}$ be an edge of
$\tau$.  Define $a_1:\tau\hookleftarrow \tau_1$ and
$a_2:\tau\hookleftarrow \tau_2$ be the two subgraphs obtained by
breaking the edge (see diagram 1).

%Without loss of generality, suppose that $E(\tau_1) = E$.  
Now the
morphism $\Kbmo{X}{a_1}:\Kbmo{X}{\tau}\rightarrow \Kbmo{X}{\tau_1}$ is
the composition of an open immersion and the projection of the fiber
product
\begin{equation}
\Kbmo{X}{\tau_1} \times_{\text{ev}_{f_1},X, \text{ev}_{f_2}}
\Kbmo{X}{\tau_2}\rightarrow \Kbmo{X}{\tau_1}.  
\end{equation}
By Proposition~\ref{prop-bbflat} the morphism $\text{ev}_{f_2}$ is
flat.  Therefore $\Kbmo{X}{a_1}$ is flat.  So $M$ dominates an
irreducible component $M_1$ of $\Kbmo{X}{a_1}$.  Since $l(\tau_1) =
l(\tau) - l(\tau_2)$ and $l(\tau_2) > 0$ by the assumption that
$E(\tau_2) = E$, we have $l(\tau_1)<l(\tau)$. So by the induction
assumption, there
exists a nice contraction $\phi_1:\rho_1\rightarrow \tau_1$ and an
irreducible component $L_1\subset \Kbmo{X}{\rho_1}$ such that
$\rho_1$ is basic and such that $L_1\subset \overline{M_1}$.  

Since $\Kbm{X}{a_1}$ is proper, there
exists an irreducible subvariety $L\subset \overline{M}$ such that
$\Kbm{X}{a_1}(L)=L_1$ and such that the fiber dimension of
$L\rightarrow L_1$ is at least the fiber dimension of
$\Kbmo{X}{a_1}$.  Up to replacing $L$ by an open subset, we may
suppose $L$ is contained in one of the locally
closed substacks $\Kbmo{X}{\rho}$.  Since $\Kbm{X}{a_1}$ maps $L$ into
$\Kbmo{X}{\rho_1}$, we must have that $\rho$ is glued from
$\rho_1$ and a graph $\rho_2$ by making an edge out of $f_1$ and
$f_2$.  Moreover $\rho_2$ must contract to $\tau_2$.  But then the
dimension of $L$ is at most 
\begin{equation}
\begin{array}{c}
\text{dim}(X,\rho_1) + \text{dim}(X,\rho_2) -
\text{dim}(X) = \\
\text{dim}(X,\rho_1) +
\lt(\text{dim}(X,\tau)-\text{dim}(X,\tau_1) \rt)
-\lt(\#\text{Edge}(\rho_2) - \#\text{Edge}(\tau_2)\rt).
\end{array}
\end{equation}
Thus we conclude that $\#\text{Edge}(\rho_2)=\#\text{Edge}(\tau_2)$,
i.e. $\rho_2=\tau_2$.  So $\psi:\rho\rightarrow \tau$ is a nice
contraction and $L\subset \Kbmo{X}{\rho}$ is an irreducible component
such that $L\subset \overline{M}$.  Moreover, $l(\rho) = l(\rho_2) =
l(\tau_2) < l(\tau)$.  By the induction assumption, there exists a
nice contraction $\phi:\sigma\rightarrow \rho$ and an irreducible
component $N\subset \Kbmo{X}{\sigma}$ such that $\sigma$ is basic and
such that $N\subset \overline{L}$.  But then
$\psi\circ\phi:\sigma\rightarrow \tau$ is nice and
$N\subset\overline{M}$, i.e. the theorem is proved for $M$.  So the
theorem is proved by induction on $E$ and $l$.  
\end{proof}  

The previous theorem suggests a strategy for proving that any given
$\Kbmo{X}{\tau}$ is irreducible:  
\begin{enumerate}
\item Determine all nice contractions $\phi:\sigma\rightarrow \tau$
such that $\sigma$ is basic.
\item Determine all irreducible components $N\subset\Kbmo{X}{\sigma}$.
\item Show that for each $N$, there is a unique irreducible component
$M(N)\subset \Kbmo{X}{\sigma}$ which contains $N$.
\item Prove that all of the putative irreducible components $M(N)$ are
actually equal.
\end{enumerate}
The first step (1) is a combinatorial problem.  
The simplest case for (2) is when $\Kbmo{X}{\sigma}$ is itself
irreducible for each basic $\sigma$.
One can try to prove (3) by a deformation theory argument; if one
proves that the general point of $M$ is a smooth point of the stack
$\Kbmo{X}{\tau}$, then it follows that there is a unique irreducible
component $M(N)$ which contains $N$.
We will prove (4) by linking up basic graphs using
\emph{almost basic graphs} (we will explain this further below).  
Although one should be able to carry out this strategy in the case of
complete intersections (and perhaps even more general varieties), in
the remainder of this paper we will restrict ourselves to
hypersurfaces $X\subset \PP^N$ with $d < \frac{N+1}{2}$.  Then the
steps above all reduce to questions regarding lines on $X$.

\section{Properties of evaluation morphisms}~\label{sec-basic}

In the last two sections we investigated when an evaluation morphism
$\text{ev}_f:\Kbmo{X}{\tau}\rightarrow X$ is flat of the expected
dimension.  In this section we also investigate when the general fiber
is irreducible, and when the morphism $\text{ev}_f$ is
\emph{unobstructed} at a general point of $\Kbmo{X}{\tau}$.  By the
same techniques in the last two sections, we reduce these properties
for a general basic $A$-graph $\tau$ to the property for $A$-graphs of
the form $\tau_1(e)$ with $e=1,\dots,E(X)$.  Using this result, we
carry out Steps $(2)$ and $(3)$ of the strategy of proof in the
previous sections.

The new property of evaluation morphisms we want to consider is the
following. 

\begin{defn}  Suppose $X\subset \PP^N$ is a smooth subvariety, $\tau$
  is a stable $A$-graph and $f\in\text{Flag}(\tau)$.   We say that
  $\cB(X,\tau,f)$ holds if we have
\begin{enumerate}
\item $\fe(X,\tau,f)$ holds,
\item the general fiber of $\text{ev}_{f}$ is geometrically
  irreducible,
\item in $\Kbmo{X}{\tau}$ there is a point $\lt[h:C\rightarrow
  X\rt]$ which is \emph{free}, i.e. $h^*T_X$ is generated by global
  sections.
\end{enumerate}
\end{defn}

The difficult item to check is still $(1)$.  We return to this point
at the end of this section.  First we give a reformulation of
item $(3)$ of the definition above.

\begin{lem}~\label{lem-Breform}
Suppose that $(C,q_{f_i})$ is a genus $0$ prestable curve with dual
graph $\tau$, and suppose that $E$ is a locally free sheaf on $C$.
The following are equivalent:
\begin{enumerate}
\item $E$ is generated by global sections.
\item For each irreducible component $C_v$ of $C$, the restriction
  $E_v$ of $E$ to $C_v$ is generated by global sections.
\item For each irreducible component $C_v$ of $C$ and each point $q\in
  C_v$, we have $H^1(C_v,E_v(-q))= 0$.
\end{enumerate}
\end{lem}

\begin{proof}
Clearly $(1)$ implies $(2)$.  By Grothendieck's Lemma~\cite[exercise
V.2.4]{H}, $E_v$ splits as a direct sum of line bundles
$L_1\oplus\dots \oplus L_r$.  If $(2)$ is satisfied, then each $L_v$
is generated by global sections, i.e. if we identify $C_v$ with
$\PP^1$, then $L_i=\OO_{\PP^1}(a_i)$ with $a_i\geq 0$.  Since
$H^1(\PP^1,\OO_{\PP^1}(a-1)) =0$ for $a\geq 0$, we conclude that
$H^1(C_v,E_v(-q)) = 0$.  So $(2)$ implies $(3)$.  

Finally suppose that $(3)$ is satisfied.  We shall prove that $E$ is
generated by global sections by induction on the number of vertices of
$\tau$, i.e. on the number of irreducible components of $C$.  If $C$
has a single irreducible component, then 
$C$ is isomorphic to $\PP^1$.  By Grothendieck's Lemma we know
$E=\OO_{\PP^1}(a_1)\oplus \dots \oplus \OO_{\PP^1}(a_r)$ for some
integers $a_i$.  Since $H^1(\PP^1,E(-q))=0$, we conclude that each
$a-1\geq -1$, i.e $a\geq 0$.  So $E$ is generated by global sections.

Now suppose that $\tau$ has more than one vertex and let $v_1$ be any leaf
of $\tau$, i.e. $v_1$ is adjacent to exactly one other vertex.
Let $i_1:C_1\rightarrow C$ be the irreducible component associated to
$v_1$, let $i_2:C_2\rightarrow C$ be the union of all the other
irreducible components of $C$ and let $q$ be the unique point of
intersection of $C_1$ and $C_2$.  Let $E_1$ denote the restriction of
$E$ to $C_1$ and let $E_2$ denote the restriction of $E$ to $C_2$.  By
the induction assumption, we may assume that $E_2$ is generated by
global sections.  But now we have an exact sequence of sheaves on $C$:
\begin{equation}
0\rightarrow (i_1)_*(E_1(-q)) \rightarrow E \rightarrow (i_2)_*(E_2)
\rightarrow 0.
\end{equation}
The obstruction to lifting the global sections of $E_2$ to global
sections of $E$ is an element of $H^1(C_1,E_1(-q))$, which is zero by
assumption.  So every global section of $E_2$ is the restriction of a
global section of $E$.  Thus the locus where $E$ isn't generated by
global sections (i.e. the cokernel of the morphism $H^0(C,E)\otimes_\CC
\OO_E\rightarrow E$) is a closed subset of $C_1 - C_2$.  Since $\tau$ has more
than one vertex, we can find a second leaf $v_2$ of $\tau$.  Repeating
the argument with $v_2$ we conclude that $E$ is generated by global
sections.
\end{proof}

\begin{lem}~\label{lem-Breform2}
With the same notation as in Lemma~\ref{lem-Breform}, if $E$ satisfies
any of the three equivalent conditions above, and if $p\in C$ is any
smooth point, then $H^1(C,E(-p))=0$.
\end{lem}

\begin{proof}
If $C$ has a single irreducible component, this follows from the
equivalent condition $(3)$ in Lemma~\ref{lem-Breform}.  
Suppose that
$C$ has $l>1$ irreducible components.  By way of induction, suppose
that the lemma has been proved for all curves with fewer than $l$
irreducible components.
We can find a leaf
$v_1$ of $C$ such that $p$ is not contained in the corresponding
irreducible component $C_2$.  
Let $C_2$, $i_1$, $i_2$, and $q$ be as in the proof of
Lemma~\ref{lem-Breform}.  Then we have a short exact sequence:
\begin{equation}
0 \rightarrow (i_1)_*(E_1(-q)) \rightarrow E(-p) \rightarrow
(i_2)_*(E_2(-p)) \rightarrow 0.
\end{equation}
By the induction assumption, both $H^1(C_1,E_1(-q))=0$ and
$H^1(C_2,E_2(-p))=0$.  So by the long exact sequence in cohomology
associated to the short exact sequence above, we conclude that
$H^1(C,E(-p))=0$.  So the lemma is proved by induction on $l$.
\end{proof}

\begin{lem}~\label{lem-Blines}
Suppose $n>2$ and $X\subset \PP^n$ is a general hypersurface of degree
$d<\frac{n+1}{2}$.  Then $\cB(X,\tau_1(1),f_1)$ holds.
\end{lem}

\begin{proof}
By Theorem~\ref{thm-thm1}, for general $X$ we have that
$\fe(X,\tau_1(1),f_1)$ holds, i.e. we have (1).  
By Lemma~\ref{lem-dfmlines}, for general
$X$ there is a free line on $X$, i.e. we have (3).  

By Theorem~\ref{lem-smlines}, for
general $X$, $F_{0,1}(X)$ is smooth.  Thus by generic smoothness, the
general fiber of $\text{ev}_{f_1}$ is smooth.  By Lemma~\ref{lem-lem1},
the general fiber of $\text{ev}_{f_1}$ is a complete intersection in
$\PP^{n-1}$ of dimension $n-d-1 > 1$.  Thus the general fiber is
geometrically 
connected by repeated application of ~\cite[corollaire 3.5,
exp. XII]{SGA2}.  Since a smooth, geometrically connected scheme is
geometrically irreducible, we have (2).
\end{proof}

The main theorem of this section is the following:

\begin{prop}~\label{prop-basic}
Suppose $X\subset \PP^N$ is a smooth subvariety which satisfies
$\cB(X,\tau_1(e),f_1)$ for $e=1,\dots, E$.  
Let $\tau$ be an $A$-graph such that $E(\tau)\leq E$.  
Then we have the following:
\begin{enumerate}
\item For each $f\in\text{Flag}(\tau)$, we have $\cB(X,\tau,f)$.
\item $\Kbmo{X}{\tau}$ is an irreducible stack.
\end{enumerate}
\end{prop}

\begin{proof}
Both statements are trivial in case $\tau$ is empty, so assume $\tau$
is nonempty.  Observe that $(1)$ implies $(2)$: given
$v\in\text{Vertex}(\tau)$, define a new $A$-graph $\tau'$ and a
combinatorial morphism $\alpha:\tau'\hookleftarrow \tau$ which
attaches 
a new
flag $f'$ to $\tau$ at $v$.  Then
$\Kbmo{X}{\alpha}:\Kbmo{X}{\tau'}\rightarrow \Kbmo{X}{\tau}$ is
smooth, surjective with geometrically irreducible fibers.  So
$\Kbmo{X}{\tau'}$ is irreducible iff $\Kbmo{X}{\tau}$ is irreducible.
By $(1)$, $\text{ev}_{f'}:\Kbmo{X}{\tau'}\rightarrow X$ is flat and the
general fiber is geometrically irreducible.  Since $X$ is irreducible,
it follows that $\Kbmo{X}{\tau'}$ is irreducible.  So it remains to
prove $(1)$.

First of all, suppose that $\beta(\tau)=0$.  Let
$\alpha:\tau\hookleftarrow \emptyset$ be the unique morphism.  Then
$\text{ev}_f$ coincides with
$\Kbmo{X}{\alpha}:\Kbmo{X}{\tau}\rightarrow \Kbmo{X}{\emptyset}=X$.
Thus $(1)$ follows from Lemma~\ref{lem-isog}.  So we are reduced to the
case that $\beta(\tau)> 0$.

We prove $(1)$ by induction on the number of vertices of $\tau$.
Suppose that $\tau$ has a single vertex, i.e. $\tau = \tau_r(e)$ for
some $r>0$.  Let $\alpha:\tau_r(e)\hookleftarrow \tau_1(e)$ be the
unique combinatorial morphism which maps
$f_1\in\text{Flag}(\tau_1(e))$ to $f\in\text{Flag}(\tau_r(e))$.  Then
$\text{ev}_f$ factors as the composition
\begin{equation}
\begin{CD}
\Kbmo{X}{\tau_r(e)} @>\Kbmo{X}{\alpha} >> \Kbmo{X}{\tau_1(e)} @>
\text{ev}_{f_1} >> X.
\end{CD}
\end{equation}
Of course $\Kbmo{X}{\alpha}$ is an open immersion into the
$(r-1)$-fold fiber product of the universal curve, so
$\Kbmo{X}{\alpha}$ is smooth with geometrically irreducible fibers.
And by $\cB$, $\text{ev}_{f_1}$ is flat and the general fiber is
geometrically irreducible.  Therefore the composition is flat and the
general fiber is geometrically irreducible, i.e. items $(1)$ and $(2)$
of condition $\cB(X,\tau_r(e),f)$ hold.  The condition that
$h^*T_X$ is generated by global sections is independent of the number
of vertices, so item $(3)$ of $\cB(X,\tau_r(e),f)$ holds as well,
i.e. $\cB(X,\tau_r(e),f)$ holds.

Now suppose that there is more than one vertex, say
$\#\text{Vertex}(\tau) = l > 1$.    
By way of assumption, suppose that $\cB(X,\sigma,f)$ has been proved for all 
$A$-graphs $\sigma$ such that $\#\text{Vertex}(\sigma) < l$.  
Let $\{f_1,f_2\}$ be
any edge and consider the subgraphs $\alpha_1:\tau\hookleftarrow
\tau_1$ and $\alpha_2:\tau\hookleftarrow \tau_2$ as in diagram $1$.
Without loss of generality, suppose that $f$ is in $\tau_1$.  Then 
$\text{ev}_f$ factors as the composition:
\begin{equation}
\begin{CD}
\Kbmo{X}{\tau} @> \Kbmo{X}{\alpha_1} >> \Kbmo{X}{\tau_1} @>
\text{ev}_f >> X.
\end{CD}
\end{equation}
Now $\Kbmo{X}{\alpha_1}$ factors as the composition of an open
immersion and the projection
\begin{equation}
\pi_1: \Kbmo{X}{\tau_1}\times_{\text{ev}_{f_1},X,\text{ev}_{f_2}}
\Kbmo{X}{\tau_2} \rightarrow \Kbmo{X}{\tau_1}.
\end{equation}
Since $\#\text{Vertex}(\tau_i) < l$ for $i=1,2$, 
the induction assumption says
that $(2)$ holds for $\text{ev}_{f_i}:\Kbmo{X}{\tau_i} \rightarrow
X$.  Since $\text{ev}_{f_1}$ is open, the general fiber of $\pi_1$
dominates the general fiber of $\text{ev}_{f_2}$.  So $\pi_1$ is flat
and the general fiber is geometrically irreducible.  Thus the same is
true of $\Kbmo{X}{\alpha_1}$.  Since $\#\text{Vertex}(\tau_1) < l$, 
$\text{ev}_{f}:\Kbmo{X}{\tau_1}\rightarrow X$ is flat and the general
fiber is geometrically irreducible.  Thus the composition is flat and
the general fiber is geometrically irreducible, i.e. items $(1)$ and
$(2)$ of $\cB(X,\tau,f)$ hold.

Finally we consider item $(3)$ of $\cB(X,\tau,f)$.  Each of the two
projections $\Kbmo{X}{\alpha_1}$ and $\Kbmo{X}{\alpha_2}$ are
dominant.  By the induction assumption, for $i=1,2$ the set of points
in $\Kbmo{X}{\tau_1}$ which parametrize stable maps with $h*T_X$
generated by global sections is an open, dense set $U_i$.  The preimage of
each $U_i$ in $\Kbmo{X}{\tau}$ is an open dense set, and the
intersection of these two open dense sets is an open dense set.  For a
point in this intersection -- using the equivalent condition $(2)$ of
Lemma~\ref{lem-Breform} -- we have that the restriction of $h^*T_X$ to
each irreducible component with vertex $v\in \tau_1$ is generated by
global sections, and also the restriction of $h^*T_X$ to each
irreducible component with vertex $v\in \tau_2$ is generated by global
sections.  So by the equivalent condition $(2)$ of
Lemma~\ref{lem-Breform}, we conclude that $h^*T_X$ is generated by
global sections.  Thus item $(3)$ of $\cB(X,\tau,f)$ is satisfied.
This completes the proof that $\cB(X,\tau,f)$ holds, and the
proposition is proved by induction.
\end{proof}

Propsition~\ref{prop-basic} simplifies Step 2 in the strategy of the last
section to checking that $\cB(X,\tau_1(e),f_1)$ holds for all
$e=1,\dots,E(X)$.  
Next we reduce Step 3 to checking that $\cB(X,\tau_1(e),f_1)$ holds
for all $e=1,\dots,E(X)$.

\begin{prop}~\label{prop-Bdefm}
Suppose that $\tau$ is a stable $A$-graph, $f\in\text{Tail}(\tau)$ and
suppose that $\cB(X,\tau,f)$ holds.
Suppose that
$\alpha:\tau\rightarrow \sigma$ is a contraction. 
The morphism $\Kbmo{X}{\alpha}$ maps a
general point of $\Kbmo{X}{\tau}$ to a point in the smooth locus of
the morphism $\text{ev}_f:\Kbm{X}{\sigma}\rightarrow X$.
\end{prop}

\begin{proof}
As in the proof of Proposition~\ref{prop-basic}, the case that
$\beta(\sigma)=0$ follows from Lemma~\ref{lem-isog}.  So we are reduced to
the case $\beta(\sigma) > 0$.  

Let $\mathfrak{M}(\sigma)$ denote the (non-separated) Artin stack of
prestable $\sigma$-curves as in~\cite[definition 2.6]{BM}.  There is a
$1$-morphism $\Kbm{X}{\sigma}\rightarrow \mathfrak{M}(\sigma)$ given
by forgetting the map to $X$.  ``Remembering'' the map to $X$ gives an
isomorphism of $\Kbm{X}{\sigma}$ with the relative scheme of morphisms
$\text{Mor}_{{\mathfrak M}}(\sigma)(\sigma,X)$, by ~\cite[prop. 4]{B}.
Since ${\mathfrak M}(\sigma)$ is smooth by ~\cite[prop. 2]{B}, to
prove that $\text{ev}_{f}$ is smooth at a point, it suffices to prove
the following morphism is smooth at this point:
\begin{equation}
(\pi,\text{ev}_f):\Kbm{X}{\sigma} \rightarrow {\mathfrak M}\times X.
\end{equation}
By ~\cite[Theorem II.1.7]{K}, to check that a point
$\zeta =\lt((C_v), (q_{f'}), (h_v)\rt)$ of $\Kbm{X}{\sigma}$ is in the
smooth locus of $(\pi,\text{ev}_f)$, it suffices to check that
$H^1(C,h^*T_X(-q_f)) = 0$.  For a general point in the image of
$\Kbmo{X}{\alpha}$, this follows from item $(3)$ of $\cB(X,\tau,f)$
along with Lemma~\ref{lem-Breform2}.
\end{proof}

\begin{cor}~\label{cor-Bdefm}
Suppose $X\subset \PP^N$ is a smooth subvariety which satisfies
$\cB(X,\tau_1(e),f_1)$ for all $e=1,\dots,E$.  Let $\tau$ be an
$A$-graph with $E(\tau) \leq E$ and suppose that
$\alpha:\tau\rightarrow \sigma$ is a contraction.  The morphism
$\Kbmo{X}{\alpha}$ maps a general point of $\Kbmo{X}{\tau}$ to a
smooth point of $\Kbm{X}{\sigma}$.
\end{cor}

\begin{proof}
This is trivial if $\tau$ is empty.  Suppose $\tau$ is not empty and
let $v$ be a vertex of $\tau$.  Let $\tau\hookleftarrow \tau'$ be the
combinatorial morphism which attaches a new tail, $f$, at the vertex
$v$.  Let $\sigma\hookleftarrow \sigma'$ be the combinatorial morphism
which attaches a new tail, $f$, at the vertex of $\sigma$ which is the
image of $v$.  Let $\alpha':\tau'\rightarrow \sigma'$ be the
contraction which restricts to $\alpha$ and which maps $f$ to $f$.  
By Proposition~\ref{prop-basic}, $\cB(X,\tau',f)$ holds.
By
Proposition~\ref{prop-Bdefm}, $\Kbmo{X}{\alpha'}$ maps a general point
of $\Kbmo{X}{\tau'}$ to a point in the smooth locus of
$\text{ev}_f:\Kbm{X}{\sigma'}\rightarrow X$.  A point in the smooth
locus of $\text{ev}_f$ is a smooth point of $\Kbm{X}{\sigma'}$.  The
image of this point in $\Kbm{X}{\sigma}$ is also a smooth point.
Since $\Kbmo{X}{\tau'}$ surjects onto $\Kbmo{X}{\tau}$, a general
point of $\Kbmo{X}{\tau'}$ maps to a general point of
$\Kbmo{X}{\tau}$.  Thus $\Kbmo{X}{\alpha}$ maps a general point of
$\Kbmo{X}{\tau}$ to a smooth point of $\Kbm{X}{\sigma}$.  
\end{proof}

\textbf{Remark:}
Now suppose $\cB(X,\tau_1(e),f_1)$ holds for all $e=1,\dots,E$,
suppose that $\tau$ is a stable $A$-graph with $E(\tau)\leq E$ and
suppose that $\alpha:\tau\rightarrow \sigma$ is a contraction.  By
Corollary~\ref{cor-Bdefm} and $(2)$ of Proposition~\ref{prop-basic},
we conclude that there is a unique irreducible component $M(\alpha)$
of $\Kbm{X}{\sigma}$ which contains the image of $\Kbmo{X}{\alpha}$,
and $M(\alpha)$ is smooth of the expected dimension at a general
point.  So Steps $(2)$ and $(3)$ of the strategy in the last section
are successful.  

Finally we give a simpler criterion for when $\cB(X,\tau,f)$ holds for all
$\tau$ with $E(\tau)\leq E$, where $E$ is some fixed integer, and also
reduce the number of components $M(\alpha)$ we have to deal with in
Step $(4)$ of our strategy.

\begin{prop}~\label{prop-simplify}
Suppose that $X\subset \PP^N$ is a smooth subvariety satisfying
\begin{enumerate}
\item $\cB(X,\tau_1(1),f_1)$ holds,
\item $\fe(X,\tau_1(e),f_1)$ holds for $e=1,\dots,E$, and
\item $\Kbmo{X}{\tau_0(e)}$ is irreducible for $e=1,\dots,E$.
\end{enumerate}
Then for each stable $A$-graph $\tau$ with $E(\tau)\leq E$ and each
flag $f\in\text{Flag}(\tau)$, $\cB(X,\tau,f)$ holds and there is a
nice contraction $\alpha:\sigma\rightarrow \tau$ such that
$E(\sigma) \leq 1$ and such that $\Kbmo{X}{\alpha}$ maps the general
point of $\Kbmo{X}{\sigma}$ to a smooth point of $\Kbm{X}{\tau}$.
\end{prop}

\begin{proof}
It is easy to see that there is always a nice contraction
$\alpha:\sigma\rightarrow \tau$ such that $E(\sigma) \leq 1$.  By
Proposition~\ref{prop-Bdefm}, we know that $\Kbmo{X}{\alpha}$ maps a
general point of $\Kbmo{X}{\sigma}$ to a smooth point of
$\Kbm{X}{\tau}$.  

By Proposition~\ref{prop-basic}, to prove that $\cB(X,\tau,f)$ 
holds for all $\tau$ with $E(\tau)\leq E$, 
it suffices to prove that $\cB(X,\tau_1(e),f_1)$ holds for all
$e=1,\dots,E$.  
Let $\Kbm{X}{\tau_1(e)}'$ denote the normalization
of $\Kbm{X}{\tau_1(e)}$.  Consider the Stein factorization
$\Kbm{X}{\tau_1(e)}'\rightarrow Z\rightarrow X$ of $\text{ev}_{f_1}$.  By
Proposition~\ref{prop-Bdefm}, there is a open, dense subset $U\subset
\Kbmo{X}{\sigma}$ such that $\Kbmo{X}{\alpha}$ maps $U$ into the
smooth locus of $\text{ev}_{f_1}$.  So $\Kbmo{X}{\alpha}|_U:U\rightarrow
\Kbm{X}{\tau_1(e)}$ factors through $\Kbm{X}{\tau_1(e)}'$.  Now consider the
image $V$ of $U$ in $Z$.  By Proposition~\ref{prop-basic}, the general
fiber of $\text{ev}_{f_1}|_U:U\rightarrow X$ is geometrically
irreducible.  Therefore $V\rightarrow X$ is generically injective.

Let $\overline{V}\subset Z$ be the Zariski closure of $V$ with the
induced, reduced scheme structure.  Then $\overline{V}$ is an
irreducible stack and $\overline{V}\rightarrow X$ is surjective and
generically injective.  In particular, $\overline{V}$ is a nonempty
irreducible component of $Z$.  So the preimage of $V$ in
$\Kbm{X}{\tau_1(e)}$ is a nonempty irreducible component of
$\Kbm{X}{\tau_1(e)}$.  By Corollary~\ref{cor-bbflat2}, every stratum in the
Behrend-Manin decomposition of $\Kbm{X}{\tau_1(e)}$ has the expected
dimension.  Thus $\Kbmo{X}{\tau_1(e)}$ is Zariski dense in
$\Kbm{X}{\tau_1(e)}$.  By assumption, $\Kbmo{X}{\tau_0(e)}$ is
irreducible.  Since $\Kbmo{X}{\tau_1(e)}\rightarrow
\Kbmo{X}{\tau_1(e)}$ is smooth with geometrically connected fibers,
also $\Kbmo{X}{\tau_1(e)}$ is irreducible.  Therefore
$\Kbm{X}{\tau_1(e)}$ is irreducible.  Therefore $\overline{V}=Z$ and
we conclude that the general fiber of $\Kbm{X}{\tau_1(e)}'\rightarrow
X$ is normal and geometrically connected, thus geometrically
irreducible.  It follows that the general fiber of
$\Kbmo{X}{\tau_1(e)}\rightarrow X$ is geometrically connected, i.e. we
have established item $(2)$ of the definition of $\cB(X,\tau_1(e),f_1)$.

To establish item $(3)$ of the definition of $\cB(X,\tau_1(e),f_1)$,
observe that the locus of points in $\Kbm{X}{\tau_1(e)}$ parametrizing
stable maps for which $h^*T_X$ is generated by global sections is an
open locus.  By Lemma~\ref{lem-Breform} and the assumption
$\cB(X,\tau_1(1),f_1)$, for a general point of $\Kbmo{X}{\sigma}$, we
have that $h^*T_X$ is generated by global sections.  So this open set
intersects the general point of the image of $\Kbmo{X}{\alpha}$, so it
is nonempty.  Therefore it intersects $\Kbmo{X}{\tau}$ and item $(3)$
follows.  
\end{proof}

We summarize the results of this section for the case of complete
intersections in the following corollary.

\begin{cor}~\label{cor-Bsum}
Suppose that $X\subset \PP^N$ is a smooth complete intersection of
threshold degree $E(X)$ which satisfies:
\begin{enumerate}
\item $\cB(X,\tau_1(1),f_1)$ holds,
\item $\fe(X,\tau_1(e),f_1)$ holds for $e=1,\dots,E(X)$, and
\item $\Kbmo{X}{\tau_0(e)}$ holds for $e=1,\dots,E(X)$.
\end{enumerate}
Then we have
\begin{enumerate}
\item For each basic $A$-graph $\tau$ and each flag
  $f\in\text{Flag}(\tau)$, $\cB(X,\tau,f)$ holds.
\item For each stable $A$-graph $\tau$ and each contraction
  $\alpha:\sigma\rightarrow \tau$ of a basic $A$-graph $\sigma$ to
  $\tau$, there is a unique irreducible component $M(\alpha)$ of
  $\Kbm{X}{\tau}$ which contains the image of $\Kbmo{X}{\alpha}$.
  Moreover $M(\alpha)$ is smooth of the expected
  dimension at a general point of the image of $\Kbmo{X}{\alpha}$.
\item $\Kbm{X}{\tau}$ is the union of the irreducible components
  $M(\alpha)$ as $\alpha:\sigma\rightarrow \tau$ ranges over nice
  contractions such that $E(\sigma)\leq 1$.  
\end{enumerate}
\end{cor}

\begin{proof}
The only new statement is $(3)$.  By Theorem~\ref{thm-spec}, we know
that each irreducible component of $\Kbm{X}{\tau}$ is one of the
irreducible components $M(\alpha')$ for a nice contraction
$\alpha':\sigma'\rightarrow \tau$ with $\sigma$ a basic $A$-graph.  Of
course we can find a nice contraction $\beta:\sigma\rightarrow
\sigma'$ such that $E(\sigma) \leq 1$.  Let $\alpha:\sigma\rightarrow
\tau$ be the composition of $\beta$ and $\alpha'$.  Then $M(\alpha')$
is an irreducible component which contains the image of
$\Kbmo{X}{\alpha}$.  So $M(\alpha') = M(\alpha)$, i.e. we have proved
$(3)$.
\end{proof}

\section{Equating Irreducible Components}

Suppose that $X\subset \PP^N$ is a complete intersection which
satisfies the hypotheses of Corollary~\ref{cor-Bsum}.  Then for each
stable $A$-graph $\tau$, we know that $\Kbm{X}{\tau}$ has the expected
dimension and is a union of irreducible components $M(\alpha)$ as
$\alpha:\sigma\rightarrow \tau$ ranges over nice contractions
with $E(\sigma)\leq 1$.  To prove that $\Kbm{X}{\tau}$ (and hence
$\Kbmo{X}{\tau}$) is irreducible, we are reduced to proving that the
irreducible components $M(\alpha)$ are all equal.

Suppose that $\cB(X,\tau_1(e),f_1)$ holds for all $e=1,\dots,E$ where
$E$ is some integer with $E\geq E(X)$. 
Fix a stable $A$-graph $\tau$ and
let $S_E(\tau)$ be the set of (isomorphism classes of) nice contractions
$\alpha:\sigma\rightarrow \tau$ with $E(\sigma)\leq E$.
Define a relation $\alpha \leq \alpha'$ if there exists a contraction
$\epsilon:\sigma \rightarrow \sigma'$ such that $\alpha=\alpha'\circ
\epsilon$.  If $\alpha\leq \alpha'$, then observe
$M(\alpha)=M(\alpha')$.  Form the equivalence relation $\cong$ on $S_E(\tau)$
generated by $\leq$.  Notice conclusion $(3)$ of
Corollary~\ref{cor-Bsum} implies that every equivalence class contains
a contraction $\alpha:\sigma\rightarrow \tau$ such that $E(\sigma)\leq
1$.  Since $M(\alpha)=M(\alpha')$ if $\alpha\cong \alpha'$, we see
that the number of irreducible components of $\Kbm{X}{\tau}$ is
bounded by the number of equivalence classes of $\cong$ on $S_E(\tau)$.  So to
prove that $\Kbm{X}{\tau}$ is irreducible, it suffices to prove that
every two elements of $S_E(\tau)$ are equivalent.

\begin{defn}~\label{defn-thresh'}
Given $X\subset \PP^N$ a smooth complete intersection, define the
\emph{modified threshold degree} of $X$ to be $E'(X) = \max(E(X),2)$.
\end{defn}

\begin{prop}~\label{prop-abasic}
Suppose that $X\subset \PP^N$ is a smooth complete intersection such
that for $e=1,\dots,E'(X)$, we have $\cB(X,\tau_1(e),f_1)$ holds.
Then for each positive integer $e$, every two elements of
$S_{E'(X)}(\tau_0(e))$ are equivalent.  In particular
$\Kbm{X}{\tau_0(e)}$ is irreducible.
\end{prop}

\begin{proof}
Recall a connected tree $\tau$ is called an \emph{path} if $\tau$ has
precisely 
one or two vertices (so no vertex has valence greater than $2$).  
The number of vertices in a \emph{path} is the
\emph{diameter} of the path.  Given any connected tree $\tau$, the
\emph{diameter} of $\tau$, $\text{diam}(\tau)$, 
is defined to be the maximum diameter of a
subgraph which is a path.  If $\alpha:\sigma\rightarrow \tau_0(e)$ is a
nice contraction, then there are at most $e$ vertices in $\sigma$.  So
the diameter of $\sigma$ is at most $e$.  Moreover, there is a unique
contraction $\alpha_e:\sigma_e\rightarrow \tau_0(e)$ with
$\text{diam}(\sigma_e)=e$.  Here $\sigma_e$ is the $A$-graph whose
underlying graph is the path of length $e$, and for each vertex $v\in
\sigma_e$ we have $\beta(v)=1$.  

To prove that any two elements in $S_{E'(X)}(\tau_0(e))$ are
equivalent, it suffices to prove that any two nice contractions
$\alpha:\sigma\rightarrow \tau$ with $E(\sigma)=1$ are equivalent.  We
will prove that for each such $\alpha:\sigma\rightarrow \tau$ with
$\text{diam}(\sigma) < e$, there is a nice contraction
$\alpha':\sigma'\rightarrow \tau$ such that $\sigma\cong \sigma'$,
$E(\sigma')=1$ and $\text{diam}(\sigma') \geq \text{diam}(\sigma)$.
From this it follows by induction that all such contractions are
equivalent to $\alpha_e:\sigma_e\rightarrow \tau$.  

Suppose that $\alpha:\sigma\rightarrow \tau$ is a nice contraction
with $E(\sigma)=1$ and $\text{diam}(\sigma) < e$.  Let
$\gamma\hookrightarrow \sigma$ be a subgraph which is a path such that
$\text{diam}(\gamma)=\text{diam}(\sigma)$.  Since $\gamma$ does not
equal $\sigma$, there exists a vertex $v_1$ of $\gamma$ such that the
valence of $v_1$ is at least $3$.  Let ${f_1,f_2}$ be an edge of
$\sigma$ not contained in $\gamma$ such that $\partial f_1 = v_1$.
Let $v_2=\partial f_2$.  Form the nice contraction $\epsilon:\sigma\rightarrow
\rho$ which contracts $v_1$ and $v_2$ to a common vertex $v$ of
$\sigma$ with $\beta(v)=2$.  The nice contraction
$\alpha:\sigma\rightarrow \tau_0(e)$ factors through a nice contraction
$\alpha_\rho:\rho\rightarrow \tau_0(e)$.

The image of $\gamma$ in $\rho$ is a path
$\gamma_\rho$ which contains $v$.  Now let $\gamma'\rightarrow \gamma_\rho$
be a contraction of a path of length $\text{diam}(\gamma)+1$ which
contracts two adjacent vertices $w_1$ and $w_2$ to $v$ (where
$\beta(w_1)=\beta(w_2)=1$).  There is a unique nice contraction
$\epsilon':\sigma'\rightarrow \rho$ such that $\gamma'$ is a path in
$\sigma'$, such that the restriction of $\epsilon'$ to $\gamma'$ is just
$\gamma'\rightarrow \rho$, such that every flag of $v$ not contained
in $\gamma_\rho$ is the image of a flag of $w_1$, and which is an
isomorphism from $\sigma'-\gamma'\rightarrow \rho-\gamma_\rho$.
Define $\alpha'=\alpha_\rho\circ \epsilon$.  Then
$\alpha':\sigma'\rightarrow \tau_0(e)$ is a nice contraction,
$E(\sigma')=1$, $\alpha\cong \alpha'$ and
$\text{diam}(\sigma')=\text{diam}(\sigma)+1$.  This proves the claim.

So by induction on the diameter of $\sigma$, every element of
$S_{E'(X)}(\tau_0(e))$ is equivalent to $\alpha_e:\sigma_e\rightarrow
\tau_0(e)$.  In particular $\Kbm{X}{\tau_0(e)}=\Kgnb{0}{0}{X}{e}$ is
irreducible.  
\end{proof}

\begin{cor}~\label{cor-abasic}
With the same hypotheses as in Proposition~\ref{prop-abasic}, for each
stable $A$-graph $\tau$ we have
\begin{enumerate}
\item $\Kbm{X}{\tau}$ is an integral, local complete intersection
  stack of the expected dimension $\text{dim}(X,\tau)$, and
  $\Kbmo{X}{\tau}$ is the unique dense stratum in the Behrend-Manin
  decomposition.  
\item For each flag $f\in \text{Flag}(\tau)$, $\cB(X,\tau,f)$ holds.
\item For each contraction $\alpha:\sigma\rightarrow \tau$,
  $\Kbm{X}{\tau}$ is smooth at the general point of the image of
  $\Kbmo{X}{\alpha}: \Kbmo{X}{\sigma} \rightarrow \Kbm{X}{\tau}$.
\end{enumerate}
\end{cor}

\begin{proof}
By Corollary~\ref{cor-bbflat}, $\fe(X,\tau_1(e),f_1)$ holds for all
integers $e>0$.  By assumption, $\cB(X,\tau_1(1),f_1)$ holds.  And by
Proposition~\ref{prop-abasic}, $\Kbmo{X}{\tau_0(e)}$ is irreducible
for each integer $e>0$.  Thus by Proposition~\ref{prop-simplify}, for
every stable $A$-graph $\tau$ and every flag $f\in\text{Flag}(\tau)$,
we have that $\cB(X,\tau,f)$ holds.  This establishes $(2)$.  

As in the proof of
Proposition~\ref{prop-basic}, $(2)$ implies that for every stable
$A$-graph $\tau$, $\Kbmo{X}{\tau}$ is irreducible of the expected
dimension.  
By a parameter count, we conclude that $\Kbmo{X}{\tau}$ is
the unique dense stratum of the Behrend-Manin decomposition of
$\Kbm{X}{\tau}$.  So $\Kbm{X}{\tau}$ is also irreducible of the
expected dimension, and generically smooth.  So $\cD(X,\tau)$ holds.
By Lemma~\ref{lem-propeq}, we conclude that $\lci(X,\tau)$ holds,
i.e. $\Kbm{X}{\tau}$ is a local complete intersection stack.  Since it
is generically smooth, and thus generically reduced, it is reduced.
So $\Kbm{X}{\tau}$ is an integral, local complete intersection stack
of the expected dimension $\text{dim}(X,\tau)$ and $\Kbmo{X}{\tau}$ is
the unique dense stratum in the Behrend-Manin decomposition.  This
establishes $(1)$.

Finally $(3)$ follows from $(1)$ and Corollary~\ref{cor-Bdefm}.
\end{proof}

Finally, we prove that a general hypersurface $X\subset \PP^n$ of
degree $d<\frac{n+1}{2}$ satisfies the hypotheses of
Proposition~\ref{prop-abasic}.  

\begin{prop}~\label{prop-final}
Suppose $n>2$, $d\leq \frac{n+1}{2}$ and suppose $X\subset \PP^n$ is a
hypersurface of degree $d$, so $E'(X)=2$.  If $\cB(X,\tau_1(1),f_1)$
holds for $X$ (recall from Lemma~\ref{lem-Blines} that
$\cB(X,\tau_1(1),f_1)$ holds for a general $X$) then also
$\cB(X,\tau_1(2),f_1)$ holds.  For such an $X$, the results of
Corollary~\ref{cor-abasic} hold.
\end{prop}

\begin{proof}
By
Corollary~\ref{cor-bbflat}, $\lci(X,\tau_1(2),f_1)$ holds.  
Since $\Kbmo{X}{\tau_0(2)}$ is the unique dense stratum in the
Behrend-Manin decomposition, to prove that $\Kbmo{X}{\tau_0(2)}$ is
irreducible, it is equivalent to prove that $\Kbm{X}{\tau_0(2)}$ is
irreducible.  
To see that $\Kbm{X}{\tau_0(2)}$ is irreducible, observe by
Theorem~\ref{thm-spec} that every
irreducible component of $\Kbm{X}{\tau}$ is of the form $M(\alpha)$
for a nice contraction $\alpha:\sigma\rightarrow \tau_0(2)$ with
$E(\alpha)=1$. But there is a unique such contraction, namely
$\alpha_2:\sigma_2\rightarrow \tau_0(2)$.  So $\Kbmo{X}{\tau_0(2)}$ is
irreducible.  

Now by property~\ref{prop-simplify}, $\cB(X,\tau_1(2),f_1)$ holds.  
\end{proof}

\bibliography{my}
\bibliographystyle{abbrv}

%\section{Bibliography} 
%
%~\bibitem[Behrend-Manin96]{BM} K. Behrend, Y. Manin, Stacks of Stable Maps and
%Gromov-Witten Invariants, Duke Math J. 85 (1996), 1-60
%
%\
%
%\bibitem[Clemens-Griffiths]{CG} H. Clemens, P. Griffiths, The intermediate
%Jacobian of a cubic three-fold, Annals of Math. 95 (1972), 281-356
%
%\
%
%\bibitem[Fulton-Pandharipande95]{FP} W. Fulton, R. Pandharipande,
%  Notes on stable  
%maps and Quantum Cohomology, Algebraic geometry -- Santa Cruz 1995,
%American Mathematical Society (1995), 45-96
%
%\
%
%\bibitem[Hartshorne77]{H} R. Hartshorne, Algebraic geometry, Springer-Verlag
%(1977) Graduate Texts in Mathematics, No. 52
%
%\
%
%\bibitem[Koll\'ar96]{K} J. Koll\'ar, Rational Curves on Algebraic Varieties,
%Springer-Verlag (1996) Ergebnisse der Math. vol 32 
%
%\
%
%\bibitem[Kontsevich95]{Kv} M. Kontsevich, Enumeration of Rational
%  Curves via Torus 
%Actions, The moduli space of curves, Birkh\"auser Boston (1995), 335-368
%
%\
%
%\bibitem[Matsumura86]{Ma} H. Matsumura, Commutative Ring Theory, Cambridge
%University Press (1986), Cambridge studies in adv. math. 8
%
%\
%
%\bibitem[Mumford70]{M} D. Mumford, Abelian varieties, Oxford University Press
%(1970), Tata Inst. of Fundamental Research Studies in Math. 5 
%
%\
%
%\bibitem[Tjurin71]{Tj} Tjurin, A. N., Geometry of the Fano surface of
%a non-singular 
%cubic $F\subset \PP^4$ and Torelli theorems for Fano surfaces and
%cubics, Izv. USSR. 35 (1971), 498-529

%\begin{thebibliography}{99}

% \bibitem{Lam} L. Lamport, {\em LaTeX, A Document Preparation System, User's
% Guide and reference manual} (Addison-Wesley Publishing Company, Reading MA,
% 1994)

% \bibitem{Fel} J. Feldman and C. Moore, Ergodic equivalence relations,
% cohomology and von Neumann algebras I, II, Trans. AMS 234 (1977) 289--324,
% 325--359

%\end{thebibliography}

\end{document}
