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\begin{document}

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

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

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

\begin{abstract}
This is the second in a sequence of papers on the geometry of spaces
of rational curves of degree $e$ on a general hypersurface $X \subset
\PP^n$ of degree $d$.  In ~\cite{HRS2} it is proved that if
$d<\frac{n+1}{2}$ then for each $e$ the space of rational curves is
irreducible, reduced and has the expected dimension.  In this paper it
is proved that if $d^2 + d + 1 \leq n$, then for each $e$ the space of
rational curves is a \emph{rationally connected} variety; in
particular it has negative Kodaira dimension.
\end{abstract}

\maketitle

\tableofcontents

\section{Statement of results}~\label{sec-results}
This is the second in a sequence of papers on the geometry of spaces
of rational curves on a general hypersurface $X \subset \PP^n$.  Let
$d = \text{deg}(X)$.  In ~\cite{HRS2} it is proved that if
$d<\frac{n+1}{2}$ then for each $e$ the space of rational curves of
degree $e$ is irreducible, reduced and has the expected dimension.
The main result of this paper is the following theorem.

\begin{thm} \label{thm-thm1}
If $X \subset \PP^n$ is a general hypersurface of degree $d$ and if $
n \geq d^2 + d + 1$ then for each integer $e \geq 1$ the stack
$\Kgnb{0,0}{X,e}$ is rationally connected.  More precisely, there
exists a morphism $f:\PP^1 \rightarrow
((\kgnb{0,0}{X,e})_{\text{fine}})_{\text{sm}}$ such that $f^*
T_{\kgnb{0,0}{X,e}}$ is ample.
\end{thm}

\begin{rmk} \label{rmk-thm1}
\begin{enumerate}
\item[(i)]
The scheme $\kgnb{0,0}{X,e}$ is the coarse moduli space of
$\Kgnb{0,0}{X,e}$, the open subset $(\kgnb{0,0}{X,e})_{\text{fine}}$
is the fine moduli locus, and the open subset
$((\kgnb{0,0}{X,e})_{\text{fine}})_{\text{sm}}$ is the smooth locus of
the fine moduli locus.
\item[(ii)]
For the cases $d=1,2$, a related and stronger theorem is proved in
~\cite[Thm. 3]{KP}; namely the coarse moduli space $\kgnb{0,0}{X,e}$
is \emph{rational}.  The proof relies on the fact that $X$ is a
homogeneous space and does not extend to the case $d \geq 3$.
Moreover, it is not clear from ~\cite{KP} that there exists a very
free rational curve in the smooth locus of the fine moduli locus of
the coarse moduli space.
\end{enumerate}
\end{rmk}

The space $\Kgnb{0,0}{X,e}$ is the \emph{Kontsevich moduli stack of
  stable maps}, which will be recalled below; it is a Deligne-Mumford
  stack containing the parameter space of smooth rational curves in
  $X$ of degree $e$ as an open substack.  A variety is
  \emph{rationally connected} if any two closed points are contained
  in the image of a morphism from $\PP^1$ to the variety.  Rationally
  connected varieties have negative Kodaira dimension, hence the
  schemes $\kgnb{0,0}{X,e}$ have negative Kodaira dimension.

\

The motivation behind Theorem~\ref{thm-thm1} is a conjectural
relationship between rational connectedness of $\Kgnb{0,0}{X,e}$ and a
theorem of Lang about rational points of varieties defined over the
function field of a surface.

\begin{thm}[Lang, ~\cite{Lang52}] \label{thm-Lang}
  Let $K$ be the function field of a surface over $\CC$ and let $X
  \subset \PP^n_K$ be a hypersurface of degree $d$.  If $d^2 \leq n$,
  then $X(K) \neq \emptyset$.
\end{thm}

There is a naive parameter count that suggests that if $d^2 \leq n$
then for $e \gg 0$ the stack $\Kgnb{0,0}{X,e}$ is rationally
connected, and if $d^2 \geq n+2$ then for $e \gg 0$ the stack
$\Kgnb{0,0}{X,e}$ is of general type; i.e., the Kodaira dimension of
$\Kgnb{0,0}{X,e}$ is determined by the same inequality as in Lang's
theorem.  This suggests Lang's theorem is related to rational
connectedness of spaces of rational curves.  In a personal
communication, A. J. de Jong has outlined an approach for proving that
a $K$-variety $X$ has a $K$-point if the stacks $\Kgnb{0,0}{X\otimes_K
\overline{K},e}$ are rationally connected, and a certain \emph{Brauer
obstruction} vanishes.  Of course Lang's original proof is simple and
direct.  But de Jong's approach could apply to classes of varieties
where Lang's proof does not apply, i.e., to varieties that are not
hypersurfaces in projective space.

\

The naive parameter count is not rigorous.  As it seems impossible to
make it rigorous, it is not recalled here.  In this paper a different
strategy is developed for proving the stacks $\Kgnb{0,0}{X,e}$ are
rationally connected, and this strategy is applied to hypersurfaces in
projective space.  This strategy should also apply to other varieties,
hence it is formulated in greater generality than strictly needed for
the case of hypersurfaces.

\subsection{The Kontsevich moduli space} \label{subsec-kv}
The most natural parameter space for rational curves of degree $e$ on
$X$ is the open subscheme of the Hilbert scheme parametrizing smooth
rational curves of degree $e$ on $X$, $R^e_X \subset
\text{Hilb}^{et+1}_X$.  For $e>1$, $R^e_X$ is not proper.  To study
the global geometry of $R^e_X$, e.g. to determine its Kodaira
dimension, it is necessary to embed it as an open subset of a proper
scheme.  The simplest choice is to take the closure $\overline{R^e_X}$
inside $\text{Hilb}^{et+1}_X$.  This is a poor choice for two reasons:
First, there is no simple characterization of the closed subschemes of
$X$ that correspond to points in $\overline{R^e_X}$.  Second, the
deformation theory of a closed subscheme of $X$ is difficult to work
with.

\

There is a better choice; one where the points of the closure have a
simple geometric meaning, and where the deformation theory is easier
to work with.  This choice is $\Kgnb{0,0}{X,e}$, or more generally
$\Kgnb{0,r}{X,e}$, the \emph{Kontsevich moduli space of degree $e$,
$r$-pointed, genus $0$ stable maps to $X$}.  This space has one
disadvantage over $\overline{R^e_X}$; namely $\Kgnb{0,r}{X,e}$ is a
\emph{Deligne-Mumford stack} rather than a scheme.  However, the
coarse moduli space $\kgnb{0,r}{X,e}$ is a projective scheme,
cf. ~\cite{Alexeev96}.  For the reader unfamiliar with stacks, most
occurrences of $\Kgnb{0,r}{X,e}$ can safely be replaced by
$(\kgnb{0,r}{X,e})_{\text{fine}}$, the \emph{fine moduli locus} of
$\kgnb{0,r}{X,e}$.

\

To be precise, $\Kgnb{0,r}{X,e}$ is the stack whose objects are
triples
\begin{equation*}
\zeta = ((p:\Sigma \rightarrow B,\sigma_1,\dots,\sigma_r),g:\Sigma
\rightarrow X),
\end{equation*}
consisting of a flat, proper family of curves $p:\Sigma
\rightarrow B$, a collection of $r$ disjoint sections $\sigma_i:B
\rightarrow \Sigma, i=1,\dots,r$ with image in the smooth locus of
$p$, and a morphism $g:\Sigma \rightarrow X$ such that for each
geometric point $b \in B$, the fiber $\Sigma_b$ is a connected,
at-worst-nodal curve of arithmetic genus $0$, the morphism
$g_b:\Sigma_b \rightarrow X\otimes_\CC \kappa(b)$ has no infinitesimal
automorphism fixing the marked points $\sigma_i(b)$, and the degree of
$(g_b)^* \OO_X(1)$ is $e$, cf. ~\cite{FP}, ~\cite{BM}.

\

A refinement of this stack used in this paper is the
\emph{Behrend-Manin stack}, $\Kbm(X,\tau)$, associated to a genus $0$
\emph{stable} $A$-\emph{graph} $\tau$, cf. ~\cite{BM}.  A genus $0$
stable $A$-graph is a \emph{tree with tails} with a degree associated
to each vertex that satisfies a certain stability condition.  A tree
is a graph that contains no cycles.  A tail, or a \emph{half-edge}, is
an edge that originates on a vertex but does not terminate on a vertex
(e.g. what one would get if one were to ``cut in half'' an edge of a
usual graph).  A \emph{degree function} is an assignment of a
nonnegative integer to each vertex of the graph.  And the
\emph{stability condition} is that no vertex both has degree $0$ and
valence less than $3$ (tails count toward the valence of a vertex).

\

Given an at-worst-nodal curve of genus $0$, $\Sigma$, a collection of
marked points on $\Sigma$, $\sigma_1,\dots,\sigma_r$, and a morphism
$g:\Sigma \rightarrow X$, there is an associated $A$-graph defined as
follows.  The tree is the dual graph of $\Sigma$; there is one vertex
for each irreducible component of $\Sigma$ and one edge for each node
of $\Sigma$.  For each marked point $\sigma_i$ of $\Sigma$ there is a
tail attached to the obvious vertex.  And the degree of a vertex is
simply the degree of $g^*\OO(1)$ on the corresponding irreducible
component.  The Behrend-Manin stack $\Kbm(X,\tau)$ (essentially)
parametrizes the closure of the locally closed substack of
$\Kgnb{0,r}{X,e}$ of stable maps whose associated $A$-graph equals
$\tau$.  For the precise definition, cf. ~\cite{BM}.

\

The boundary of the stack $\Kbm(X,\tau)$ is a union of stacks
$\Kbm(X,\sigma)$ where $\sigma$ ranges over stable $A$-graphs such
that there is a \emph{contraction} from $\sigma$ to $\tau$, i.e., the
graph $\sigma$ is ``more degenerate'' than the graph $\tau$, cf.
~\cite{BM}.  Hence the boundary of every Behrend-Manin stack can be
understood inductively starting from the ``most degenerate'' graphs
$\tau$, for which $\Kbm(X,\tau)$ is a fiber product over $X$ of the
space of pointed lines on $X$.  So questions about $\Kbm(X,\tau)$ that
can be studied by specializing to points in the boundary eventually
reduce to questions about the space of pointed lines on $X$.
Moreover, the deformation theory of a point in $\Kbm(X,\tau)$ is
straightforward; it will be recalled in Section ~\ref{sec-def}.

\subsection{Sketch of the proof} \label{subsec-sketch}

The proof of Theorem ~\ref{thm-thm1} uses a theorem of Koll\'ar.

\begin{thm}[Koll\'ar, Thm. IV.3.7 ~\cite{K}] \label{thm-Kollar}
  Let $V$ be an irreducible, projective variety, and let
  $V_{\text{sm}} \subset V$ denote the smooth locus of $V$.  If there
  exists a \emph{very free} morphism $f:\PP^1 \rightarrow
  V_{\text{sm}}$, i.e. a morphism such that $f^* T_V$ is an ample
  vector bundle, then $V$ is rationally connected.
\end{thm}

The reader is warned that ``very free'' is the first of a multitude of
definitions with ``very'' similar names: free, very free, deformation
ample, very stable, unobstructed, twisting, very twisting, twistable,
very twistable, positive, very positive, inducting pair, inductable,
modification, typical, and $c$-generating linear system.  A morphism
$f:\PP^1 \rightarrow V$ is \emph{free} if $f^* T_V$ is a vector bundle
that is generated by global sections.  The goal is to prove there
exists a very free morphism to $\Kgnb{0,0}{X,e}$ for all $e$.  It is
difficult to directly construct a very free morphism.  However,
existence of a very free morphism can be studied by specializing to
the boundary of $\Kgnb{0,0}{X,e}$, and, using an induction argument,
can ultimately be reduced to a question about pointed lines on $X$.
The induction argument does not produce a very free morphism to the
boundary of $\Kgnb{0,0}{X,e}$, but rather a reducible rational curve
$B$ in the boundary such that the restriction of $T_{\Kgnb{0,0}{X,e}}$
to $B$ is \emph{deformation ample}: for a deformation of $B$ to an
irreducible rational curve $B'$, the restriction of
$T_{\Kgnb{0,0}{X,e}}$ to $B'$ is \emph{ample}, cf.
Section~\ref{sec-DA} (a curve $B'$ is called a \emph{deformation} of
$B$ if both $B$ and $B'$ have a common generization).  The rational
curve $B'$ is a very free rational curve which proves that
$\Kgnb{0,0}{X,e}$ is rationally connected.

\

The induction argument constructs a reducible rational curve $B$ which
itself parametrizes stable maps from reducible rational curves to $X$.
Each of these stable maps is the union of a map of degree $e-1$,
$g:\Sigma\rightarrow X$, and a line $L \subset X$ that intersect in a
node $\sigma \in \Sigma \cap L$.  Since the union is stable, the
$1$-pointed map $(\Sigma,\sigma,g)$ is stable.  To simplify the
deformation theory, it is assumed that $(\Sigma, \sigma,g)$ is
\emph{very stable}: the unmarked map $(\Sigma,g)$ is stable (this
assumption must be justified!).  To guarantee that $B$ is in the
smooth locus of $\Kgnb{0,0}{X,e}$, it is assumed that $(\Sigma,g)$ is
\emph{unobstructed}: $\EExt{\Sigma}^1(L_g, \OO_{\Sigma}) =\{0 \}$.

\

The condition that $T_{\Kgnb{0,0}{X,e}}|_B$ is deformation ample can
be translated into a condition on the family of pointed lines,
$(L,\sigma)$, together with a condition on the family of pointed maps
of degree $e-1$, $(\Sigma,\sigma,g)$; these conditions are defined in
Section ~\ref{sec-props}.  The condition on the family of pointed
lines is that it is very twisting.  The family is \emph{twisting}
(resp. \emph{very twisting}) if the associated morphism $\zeta:B
\rightarrow \Kgnb{0,1}{X,1}$ pulls back the vertical tangent bundle of
the projection $\text{pr}:\Kgnb{0,1}{X,1} \rightarrow \Kgnb{0,0}{X,1}$
to a bundle which is generated by global sections, and pulls back the
vertical tangent bundle of the ``evaluation at the marked point''
morphism $\text{ev}:\Kgnb{0,1}{X,1} \rightarrow X$ to a bundle which
is generated by global sections (resp. deformation ample).  Composing
$\zeta$ with $\text{ev}$ gives a map $h:B \rightarrow X$.  Such a map
which arises from a twisting (resp. very twisting) family is called
\emph{twistable} (resp. \emph{very twistable}).  The property of being
twistable (resp. very twistable) is an open condition on the family of
all maps.

\

The condition on the family of pointed maps of degree $e-1$ is that it
is very positive.  The family is \emph{positive} (resp. \emph{very
positive}) if the associated morphism $\zeta:B \rightarrow
\Kgnb{0,1}{X,e-1}$ pulls back the vertical tangent bundle of the
projection $\text{pr}:\Kgnb{0,1}{X,1} \rightarrow \Kgnb{0,0}{X,1}$ to
a bundle which is generated by global sections (resp. ample) and the
pullback by $\text{pr}\circ \zeta$ of $T_{\Kgnb{0,0}{X,1}}$ is
deformation ample.  The main observation is this: For a positive
(resp. very positive) family over a smooth rational curve, the
morphism $\zeta:B \rightarrow \Kgnb{0,1}{X,e-1}$ is free (resp. very
free).

\

An \emph{inducting pair} consists of a very twisting family of pointed
lines over $B$ and a very positive family of pointed maps of degree
$e-1$ over $B$ which intersect along the marked points.  The induction
step proves that if an inducting pair exists for degree $e$, then an
inducting pair exists for degree $e+1$.  It is sometimes useful to
``forget'' the very twisting family of pointed lines, and only
``remember'' that the family of marked points is a very twistable map.
A very positive family of pointed maps of degree $e-1$ such that the
family of marked points is a very twistable map is an
\emph{inductable} family.

\

The induction step begins with an inducting pair for degree $e$ and
produces an inductable family for degree $e+1$.  In particular, the
inductable family is very positive and there is a deformation to a
very free morphism from an irreducible curve to $\Kgnb{0,1}{X,e}$
whose projection to $\Kgnb{0,0}{X,e}$ is a very free morphism.  The
family of unmarked maps over $B$ is obtained by gluing the very
twisting family of lines and the very positive family of maps of
degree $e-1$ along the curve of marked points $\sigma$.  To get a
family of marked maps, let $\sigma'$ be an irreducible curve in the
total space $\Sigma$ of the very positive family which is linearly
equivalent to $\sigma$.  The curve $\sigma'$ fails to be a family of
marked points precisely at the finitely many points in $B$ over which
the curve $\sigma'$ intersects the curve $\sigma$.  The solution is to
blow up each of these finitely many points on $\Sigma$; this produces
a family of marked stable maps of degree $e$.  Unfortunately blowing
up destroys the ``very positivity'' of the family.  Very positivity is
restored by making a \emph{modification} at each of the finitely many
points of $B$ over which $\Sigma$ is blown up.  The modification
attaches a $\PP^1$ to $B$ at the specified point and extends the
family over this $\PP^1$ so that the stabilized family of unmarked
maps is a constant family, but such that the marked points in this
constant family vary.  After modification, the family of marked stable
maps is inductable.  Incidentally, it is the process of modification
that requires using reducible curves in $\Kgnb{0,0}{X,e}$ and working
with the property of deformation ample bundles.

\

The last section establishes the base case of the induction argument:
existence of a family of pointed lines that is both very twisting and
very positive.  The total space of a very positive family of lines is
a scroll $\Sigma$.  The whole argument reduces to producing a pair of
a scroll $\Sigma$ and a hypersurface $X$ containing $\Sigma$ such that
$\Sigma$ corresponds to a very positive family of lines on $X$.  This
reduces to a computation of the dimension of certain linear systems on
the scroll $\Sigma$, $c$-\emph{generating linear systems}.  This is a
straightforward computation in the Cox homogeneous coordinate ring of
the scroll.  The hypothesis that $d^2 +d + 1 \leq n$ is used in this
last computation.

\subsection{Technical Hypotheses} \label{subsec-hyp}
In the proof of the induction step, some technical hypotheses are
used.

\begin{hyp}\label{hyp-1}
  For each contraction of genus $0$ stable $A$-graphs,
  $\phi:\sigma\rightarrow \tau$, the image of the morphism of
  Behrend-Manin stacks $\Kbm(X,\sigma) \rightarrow \Kbm(X,\tau)$ has
  codimension $\text{dim}(X,\tau) - \text{dim}(X,\sigma)$ in
  $\Kbm(X,\tau)$.
\end{hyp}

By \cite[Prop. 7.4]{HRS2}, if $d < \frac{n+1}{2}$ and if $X\subset
\PP^n$ is a general hypersurface of degree $d$, each stack
$\Kbm(X,\sigma)$ has the expected dimension; thus
Hypothesis~\ref{hyp-1} holds for $X$.

\begin{hyp}\label{hyp-1.5}
  A general fiber of the evaluation map $\text{ev}:\Kgnb{0,1}{X,1}
  \rightarrow X$ is irreducible.
\end{hyp}

For a pair $(X,p)$ consisting of a hypersurface $X\subset \PP^n$ of
degree $d$ and a point $p\in X$, the associated fiber of $\text{ev}$
is a subvariety $Z\subset \PP^{n-1}$ which is a complete intersection
of a sequence of hypersurfaces $Y_1,\dots, Y_d$ in $\PP^n$ with
$\text{deg}(Y_i)=i$: the defining equation of $Y_i$ is the degree $i$
homogeneous part of the Taylor expansion of the defining equation of
$X$ about the point $p$.  If the pair $(X,p)$ is general, the sequence
of hypersurfaces $Y_1,\dots,Y_d$ is general.  By the Bertini theorem
~\cite[Thms. 4.10, 6.10]{Jou} the intersection $Y_1 \cap \dots \cap
Y_d$ is smooth and connected if $d \leq n-2$.

\begin{hyp}\label{hyp-1.75}
  For each integer $e\geq 0$, the locus in $\Kgnb{0,1}{X,e}$
  parametrizing stable maps with nontrivial automorphism group has
  codimension at least $2$.
\end{hyp}

Of course any stable map with nontrivial automorphism group has an
irreducible component which is a multiple cover of its image.  In
light of \cite[Prop. 7.4]{HRS2}, a simple parameter count shows that
if $d\leq \frac{n+1}{2}$ and if $X \subset \PP^n$ is a general
hypersurface of degree $d$, Hypothesis~\ref{hyp-1.75} is satisfied.

\subsection{Conventions} \label{subsec-conv}
Unless stated otherwise, schemes are of finite type and separated over
$\SP \CC$.  Absolute fiber products of schemes will be fiber products
over $\SP \CC$.  Absolute fiber products of stacks will be $2$-fibered
products over $\SP \CC$.

\subsection{Acknowledgments} \label{subsec-ack}
We are very grateful to A. J. de Jong and Steven Kleiman for many
useful conversations.  We are also grateful to the referees for their
many useful comments.

\section{Deformation ample}~\label{sec-DA}

Let $T$ be a scheme.

\begin{defn} \label{defn-prestab0}
A \emph{family of prestable curves of genus $g$ over $T$} is a proper,
flat morphism $\pi:B \rightarrow T$ such that every geometric fiber of
$\pi$ is a connected, at-worst-nodal curve of arithmetic genus $g$.
\end{defn}

\begin{notat} \label{notat-mu}
Let $\pi:B\rightarrow T$ be a morphism of schemes and let $E$ be a
quasi-coherent sheaf on $B$.  Denote by $\mu_E:\pi^*\pi_* E
\rightarrow E$ the morphism left adjoint to the identity morphism
$\pi_*E \rightarrow \pi_*E$.  The sheaf $E$ is \emph{$\pi$-relatively
generated by global sections} if $\mu_E$ is surjective.
\end{notat}  

Let $\pi:B\rightarrow T$ be a family of prestable curves of genus $0$ 
and let $E$ be a
coherent sheaf on $B$.  Let $\sigma:T \rightarrow B$ be a section of
$\pi$ and let ${\mc I} \subset \OO_B$ be the ideal sheaf of
$\sigma(T)$.

\begin{lem}~\label{lem-gend}
\begin{enumerate}
\item[(i)] 
If $E$ is $\pi$-relatively generated by global sections, then
  $R^1\pi_* E = \{0\}$.
\item[(ii)]
If $E$ is $\pi$-relatively generated by global sections, then
$R^1\pi_*(\mc{I}\cdot E) = \{0\}$.
\item[(iii)]
Let $E'$ and $E''$ be coherent sheaves on $B$ and let
\begin{equation*}
\begin{CD}
0 @>>> E' @>>> E @>>> E'' @>>> 0
\end{CD}
\end{equation*}
be a short exact sequence of coherent sheaves.  If $E'$ and $E''$ are
$\pi$-relatively generated by global sections, then $E$ is
$\pi$-relatively generated by global sections.
\end{enumerate}
\end{lem}

\begin{proof}
All three statements can be proved locally over $T$.  Hence it
suffices to consider the case that $T$ is affine.

\textbf{(i)}: Since $E$ is $\pi$-relatively generated by global
  sections and $T$ is affine, $E$ is generated by global sections,
  i.e., there is a short exact sequence of coherent sheaves,
\begin{equation*}
\begin{CD}
0 @>>> K @>>> \OO_B^{\oplus N} @>>> E @>>> 0.
\end{CD}
\end{equation*}
Since $\pi$ is of relative dimension $1$, $R^2\pi_*K=\{0\}$.  Since
the fibers of $B$ are connected of arithmetic genus $0$, $R^1\pi_*
\OO_B=\{0\}$.  In the long exact sequence of higher direct images
associated to the short exact sequence above, $R^1\pi_*E$ fits between
$R^1\pi_*\OO_B^{\oplus N}$ and $R^2\pi_*K$; hence $R^1\pi_*E=\{0\}$.

\

\textbf{(ii)}: There is a short exact sequence of coherent sheaves,
\begin{equation*}
\begin{CD}
0 @>>> {\mc I}\cdot E @>>> E @>>> E\otimes_{\OO_B} \OO_{\sigma(T)} @>>> 0
\end{CD}
\end{equation*}
giving rise to a long exact sequence of cohomology groups,
\begin{equation*}
\begin{CD}
\pi_*E @>>> \pi_*(E\otimes_{\OO_B} \OO_{\sigma(T)}) 
@>>> R^1\pi_*({\mc I}_b\cdot E) @>>> R^1\pi_*E.
\end{CD}
\end{equation*}
By the last paragraph, $R^1\pi_*E = \{0\}$.  Since $E$ is generated by
global sections, $\pi_*E \rightarrow \pi_*(E \otimes_{\OO_B}
\OO_{\sigma(T)})$ is surjective.  Therefore $R^1\pi_*({\mc I} \cdot E)
=\{0\}$.

\

\textbf{(iii)}: By (i), $h^1(B,E')=0$; therefore every global section
of $E''$ is the image of a global section of $E$.  So the global
sections of $E$ generate $E''$.  And the global sections of $E'$
generate $E'$.  Therefore $E$ is generated by global sections.
\end{proof}

Let $B$ be a prestable curve of genus $0$, and let $E$ be a locally
free sheaf of positive rank on $B$.  A \emph{smoothing} of the pair
$(B,E)$ over a discrete valuation ring $R$ is a pair $({\mc B},{\mc
E})$ consisting of a family $\mc{B} \rightarrow \SP R$ of prestable
curves of genus $0$ and a locally free sheaf $\mc{E}$ such that the
generic fiber of $\mc{B}$ is a smooth curve, such that the closed
fiber of $\mc{B}$ is isomorphic to $B$, and such that the restriction
of $\mc{E}$ to the closed fiber is isomorphic to $E$.  What conditions
on $(B,E)$ guarantee that for every smoothing $({\mc B},{\mc E})$, the
restriction of ${\mc E}$ to the generic fiber is an ample locally free
sheaf? Certainly if $E$ is ample, this is true.  But $E$ need not be
ample for this condition to hold: e.g. if $E$ is an invertible sheaf
such that the total degree of $E$ is positive, then for every
smoothing the restriction of $\mc{E}$ to the generic fiber is ample.
Although it is not the most general criterion, the following criterion
is used in the rest of the paper.

\

\begin{defn}~\label{def-DA}
  Let $B$ be a connected, proper, at-worst-nodal curve of arithmetic
  genus $0$.  A locally free sheaf $E$ on $B$ with positive rank is
  \emph{deformation ample} if
  \begin{enumerate}
    \item[(i)] 
     $E$ is generated by global sections, and
    \item[(ii)] 
     $h^1(B,E(K_B))=0$, where $\OO_B(K_B)$ is the dualizing
      sheaf of $B$.
  \end{enumerate}
\end{defn}

\begin{rmk}~\label{rmk-DA}
  \begin{enumerate}
    \item[(i)] Conditions (i) and (ii) above are independent.
    \item[(ii)] If $E$ is invertible, then $E$ is deformation ample
      iff the restriction of $E$ to every irreducible component has
      nonnegative degree and the restriction to at least one
      irreducible component has positive degree, cf. Lemma
      ~\ref{lem-DAcrit}.
    \item[(iii)] One can determine whether $E$ is deformation ample in
      terms of the splitting type of the restriction of $E$ to each
      irreducible component together with the patching isomorphisms at
      the nodes of $B$.
  \end{enumerate}
\end{rmk}

Let $T$ be a scheme, let $\pi:B \rightarrow T$ be a family of
prestable curves of genus $0$, and let $E$ be a locally free sheaf of
positive rank on $B$.

\begin{defn}~\label{defn-fDA}
  The sheaf $E$ is \emph{$\pi$-relatively deformation ample} (or
  simply \emph{deformation ample} if $\pi$ is understood) if
  \begin{enumerate}

    \item[(i)] $E$ is $\pi$-relatively generated by global sections,
and

    \item[(ii)] $R^1\pi_*(E(K_\pi)) = \{0\}$, where $\OO_B(K_\pi)$ is
the relative dualizing sheaf of $\pi$.

  \end{enumerate}
\end{defn}

Let $\pi:B \rightarrow T$ be a proper, flat family of connected,
at-worst-nodal curves of arithmetic genus $0$, let $E$ be a locally
free sheaf on $B$ of positive rank, and let $f:T' \rightarrow T$ be a
morphism of schemes.  Denote the fiber product as in the diagram,
\begin{equation*}
\begin{CD}
B' @> g >> B \\
@V \pi' VV @VV \pi V \\
T' @> f >> T
\end{CD}
\end{equation*}
Denote by $E'$ the pullback $g^* E$.

\begin{lem}~\label{lem-bcDA}
  If $E$ is $\pi$-relatively deformation ample, then $E'$ is
  $\pi'$-relatively deformation ample.  If $f$ is surjective, the
  converse also holds.
\end{lem}

\begin{proof}
\textbf{($\Rightarrow$)}: For the main direction, by
~\cite[Sec. 8.5.2, Prop.  8.9.1]{EGA4}, it suffices to consider the
case when $T$ and $T'$ are Noetherian affine schemes.

\

There is a canonical map of $\OO_{T'}$-modules, $\nu:f^*\pi_*
E\rightarrow (\pi')_*g^* E$, which fits into a commutative diagram,
\begin{equation*}
\begin{CD}
(\pi')^*f^*\pi_* E @> = >> g^* \pi^* \pi_* E \\
@V (\pi')^*\nu VV @VV g^* \mu_{E} V \\
(\pi')^*(\pi')_* E' @> \mu_{E'} >> E'
\end{CD}
\end{equation*}
Since $\mu_E$ is surjective, $g^*\mu_E$ is surjective.  Hence also
$\mu_{E'}$ is surjective, i.e., $E'$ is $\pi'$-relatively generated by
global sections.
 
\

Since $\pi$ has relative dimension $1$, $R^2\pi_* E(K_\pi) = \{0\}$.
By \cite[Thm.  III.12.11(b)]{H}, for every closed point $t\in T$,
$h^1(B_t,E(K_\pi)|_{B_t}) = 0$.  By ~\cite[Prop. III.9.3]{H}, for
every closed point $t'\in T'$, $h^1(B'_{t'},E'(K_{\pi'})|_{B'_{t'}}) =
0$.  So by \cite[Thm.  III.12.11(a)]{H} and Nakayama's lemma,
$R^1\pi'_*(E'(K_{\pi'})) = \{ 0 \}$.  Hence $E'$ is $\pi'$-relatively
deformation ample.

\

\textbf{($\Leftarrow$)}: Now suppose that $f$ is surjective and that
$E'$ is $\pi'$-relatively deformation ample.  As above, it suffices to
consider the case when $T$ and $T'$ are Noetherian affine schemes.  As
above, for every closed point $t'\in T'$,
$h^1(B'_{t'},E'(K_{\pi'})|_{B'_{t'}}) = 0$.  Since $T'\rightarrow T$
is surjective, by ~\cite[Prop. III.9.3]{H} for every closed point
$t\in T$, $h^1(B_t,E(K_\pi)|_{B_t}) = 0$.  So by ~\cite[Thm
III.12.11(a)]{H} and Nakayama's lemma, $R^1\pi_*(E(K_\pi)) = \{0\}$.

\

It remains to prove that $E$ is $\pi$-relatively generated by global
sections.  For every closed point $t\in T$, there is a closed point
$t'\in T'$ mapping to $t$.  Since $E'|_{B'_{t'}}$ is generated by
global sections, also $E|_{B_t}$ is generated by global sections.  By
Lemma~\ref{lem-gend}, $h^1\lt( B_t, E|_{B_t} \rt) =0$.  By
~\cite[Thm. III.12.11(a)]{H} and Nakayama's lemma, $R^1\pi_*(E) =
\{0\}$.

\

The claim is that for any coherent $\OO_T$-module $\mc{F}$, $R^1\pi_*(
\pi^* \mc{F} \otimes E) = \{0\}$.  This is local on $T$.  Locally on
$T$, $\mc{F}$ is the cokernel of $\OO_T^{\oplus N}$ for some $N$, so
$\pi^* \mc{F} \otimes E$ is the cokernel of $E^{\oplus N}$.  Since
$\pi$ has relative dimension $1$, $R^1\pi_*$ is right exact on the
category of coherent $\OO_B$-modules.  Since $R^1\pi_*(E^{\oplus N}) =
\{0\}$, also $R^1\pi_*(\pi^* \mc{F}\otimes E) = \{0\}$, which proves
the claim.

\

In particular, applying the long exact sequence of higher direct
images to the short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> \pi^*{\mc I}_t\otimes E @>>> E @>>> E|_{B_t} @>>> 0,
\end{CD}
\end{equation*}
$\pi_*(E) \rightarrow H^0\lt( B_t, E|_{B_t} \rt)$ is surjective.
Since $E|_{B_t}$ is generated by global sections for every closed
point $t\in T$, $E$ is $\pi$-relatively generated by global sections.
So $E$ is $\pi$-relatively deformation ample.
\end{proof}

\begin{lem}\label{lem-smDA}
  Let $\pi:B \rightarrow T$ be a proper, flat family of connected,
  \emph{smooth} curves of genus $0$ and let $E$ be a locally free
  sheaf of positive rank.  Then $E$ is $\pi$-relatively deformation
  ample iff $E$ is $\pi$-relatively ample.
\end{lem}

\begin{proof}
  Both properties are local on $T$ and can be checked after \'{e}tale,
  surjective base-change of $T$.  So it suffices to consider the case
  when $\pi:B\rightarrow T$ is isomorphic to $\pi_T:T\times \PP^1
  \rightarrow T$.

\

\textbf{($\Rightarrow$)}: Denote $F = (\pi_T)_*( E \otimes
\pi_{\PP^1}^* \OO_{\PP^1}(-1) )$.  Tensoring the map $\mu_{E\otimes
\pi_{\PP^1}^* \OO_{\PP^1}(-1)}$ with the identity map on $\pi^*
\OO_{\PP^1}(-1)$ gives a map $\nu: \pi_T^*F \otimes \pi_{\PP^1}^*
\OO_{\PP^1}(1) \rightarrow E$.  Assume that $E$ is deformation ample.
The claim is that $\nu$ is surjective.  To prove this, it suffices to
prove,
\begin{enumerate}

\item[(i)] 
For every geometric point $t$ of $T$, $h^1(\PP^1_{\kappa(t)}, E|_{B_t}
  \otimes \OO_{\PP^1}(-1)) = 0$,

\item[(ii)] 
$F\otimes_{\OO_T} \kappa(t) = H^0(\PP^1_{\kappa(t)},E|_{B_t} \otimes
  \OO_{\PP^1}(-1))$, and

\item[(iii)] 
the map $H^0(\PP^1_{\kappa(t)}, E|_{B_t}\otimes
  \OO_{\PP^1}(-1))\otimes \OO_{\PP^1}(1) \rightarrow E|_{B_t}$ is
  surjective.

\end{enumerate}

\

By Grothendieck's lemma ~\cite[Exer. V.2.6]{H}, $E|_{B_t}$ splits as a
direct sum $\OO_{\PP^1}(a_1)\oplus \dots\oplus \OO_{\PP^1}(a_r)$ for
some integers $a_1\leq \dots \leq a_r$.  By Lemma~\ref{lem-bcDA},
$E|_{B_t}$ is deformation ample, and in particular
$h^1(\PP^1,E|_{B_t}(-2)) = 0$.  Hence $a_1\geq 1$, and $h^1(\PP^1,
E|_{B_t}\otimes \OO_{\PP^1}(-1))=0$, i.e., (i) holds.  By ~\cite[Thm.
III.12.11(b)]{H}, also (ii) holds.  Finally, for $a_i \geq 1$,
$H^0(\PP^1_\kappa, \OO_{\PP^1}(a_i-1))\otimes
\OO_{\PP^1}(1)\rightarrow \OO_{\PP^1}(a_i)$ is surjective.  Thus (iii)
holds and the claim is proved.

\

Now $\pi_T^*F\otimes \pi_{\PP^1}^*\OO_{\PP^1}(1)$ is
$\pi_T$-relatively ample.  Since $E$ is a quotient of $\pi_T^*F
\otimes \pi_{\PP^1}^* \OO_{\PP^1}(1)$, also $E$ is $\pi_T$-relatively
ample, cf. Lemma~\ref{lem-secDA} (i).

\

\textbf{($\Leftarrow$)}: The converse direction follows in the same
way.
\end{proof}

\begin{lem}~\label{lem-opDA}
  There exists an open subscheme $i:U\rightarrow T$ with the following
  property: for every morphism $f:T'\rightarrow T$, $f(T')$ is
  contained in $U$ iff $E'$ is $\pi'$-relatively deformation ample.
\end{lem}

\begin{proof}
  By \cite[Sec. 8.5.2, Prop. 8.9.1]{EGA4}, it suffices to consider the
  case that $T$ and $T'$ are Noetherian affine schemes.

\

Let $Z_1\subset T$ be the closed subset,
\begin{equation*}
Z_1 = f [ \text{Supp}( \text{coker}( \pi^* \pi_* E \rightarrow E) ) ].
\end{equation*}
Let $Z_2\subset T$ be the closed subset,
\begin{equation*}
Z_2 = \text{Supp}( R^1\pi_*(E(K_\pi)) ).  
\end{equation*}
And let $Z_3 \subset T$ be the closed subset,
\begin{equation*}
Z_3 = \text{Supp}( R^1\pi_*E ).
\end{equation*}
Let $i:U\rightarrow T$ be the open complement of $Z_1\cup Z_2 \cup
Z_3$.

\

Let $f:T'\rightarrow T$ be a morphism of Noetherian affine schemes.
By ~\cite[Thm. III.12.11, Prop. III.9.3]{H} and Nakayama's lemma,
$R^1\pi'_*(E'(K_{\pi'})) = \{0\}$ iff for each closed point $t'\in
T'$, $h^1(B'_{t'},E'(K_{\pi'})|_{B'_{t'}}) = 0$.  Denoting $t=f(t')$,
$h^1(B'_{t'},E'(K_{\pi'})|_{B'_{t'}})=0$ iff
$h^1(B_t,E(K_{\pi})|_{B_t}) = 0$, i.e., iff $t$ is contained in the
complement of $Z_2$.  Hence $R^1\pi'_*(E'(K_{\pi'}))=\{0\}$ iff $f(T)$
is contained in the complement of $Z_2$.

\

By the same argument as in the converse direction of the proof of
Lemma~\ref{lem-bcDA}, $E'$ is $\pi'$-relatively generated by global
sections iff for every closed point $t'\in T'$, $E'|_{B'_{t'}}$ is
generated by global sections.  Denoting $t=f(t')$, $E'|_{B'_{t'}}$ is
generated by global sections iff $E|_{B_t}$ is generated by global
sections.  If $E|_{B_t}$ is generated by global sections, then
$h^1(B_t,E|_{B_t}) = 0$.  By ~\cite[Thm. III.12.11]{H}, $t$ is not in
$Z_3$ and $t$ is not in $Z_1$.  Conversely, if $t$ is not in $Z_3$,
then $E|_{B_t}$ is generated by global sections iff $t$ is not in
$Z_1$.  Thus $E'$ is $\pi'$-relatively generated by global sections
iff $f(T')$ is contained in the complement of $Z_1 \cup Z_3$.  So $E'$
is $\pi'$-relatively deformation ample iff $f(T')$ is contained in
$U$.
\end{proof}

\begin{lem}~\label{lem-secDA} 
\begin{enumerate}

\item[(i)] 
If $\chi: E\rightarrow E''$ is a morphism of locally free sheaves on
$B$ whose cokernel is torsion in every fiber (in particular, if $\chi$
is surjective), if $E''$ is nonzero, and if $E$ is $\pi$-relatively
deformation ample, then also $E''$ is $\pi$-relatively deformation
ample.

\item[(ii)] 
If $E'$ and $E''$ are $\pi$-relatively deformation ample, then for
every short exact sequence of coherent $\OO_B$-modules,
\begin{equation*}
\begin{CD}
0 @>>> E' @> \upsilon >> E @> \chi >> E'' @>>> 0, 
\end{CD}
\end{equation*}
$E$
is $\pi$-relatively deformation ample.

\item[(iii)] 
If $E$ is $\pi$-relatively deformation ample, then for every integer
  $n\geq 1$, also $E^{\otimes n}$ is $\pi$-relatively deformation
  ample.

\end{enumerate}
\end{lem}

\begin{proof}
\textbf{(i)}: Let $Q$ denote the cokernel of $\chi$ and let $I \subset
 E''$ denote the image of $\chi$.  There is a short exact sequence of
 coherent $\OO_B$-modules:
\begin{equation*}
\begin{CD}
0 @>>> I @>>> E'' @>>> Q @>>> 0.
\end{CD}
\end{equation*}
When we ``twist'' this exact sequence by $\OO_B(K_\pi)$, it remains
exact.  Because $\pi$ has relative dimension $1$, $R^1\pi_*$ is right
exact on the category of coherent $\OO_B$-modules.  In particular,
since $R^1\pi_* E(K_{\pi}) = \{0\}$, also $R^1 \pi_* I(K_{\pi}) =
\{0\}$.  Since $Q$ is torsion in every fiber, $R^1 \pi_* Q(K_\pi) = \{
0 \}$.  Thus, by the long exact sequence of higher direct images
associated to the twisted exact sequence above, $R^1 \pi_*
E''(K_{\pi}) = \{ 0 \}$.

\

The surjective composition map
\begin{equation*}
\begin{CD}
\pi^*\pi_*E_1 @> \mu_E >> E_1 @>>> I
\end{CD}
\end{equation*}
factors through the natural map $\mu_I:\pi^*\pi_* I\rightarrow I$.
Hence $\mu_I$ is surjective, i.e., $I$ is $\pi$-relatively generated
by global sections.  Since $Q$ is torsion in every fiber, the support
of $Q$ is finite over $T$ and it follows that $Q$ is $\pi$-relatively
generated by global sections.  By Lemma~\ref{lem-gend} (iii), $E''$ is
$\pi$-relatively generated by global sections.  So $E''$ is
$\pi$-relatively deformation ample.

\

\textbf{(ii)}: By hypothesis, $R^1 \pi_* E'(K_\pi) = R^1 \pi_*
E''(K_\pi) = \{0\}$.  By the long exact sequence of higher direct
images, also $R^1 \pi_* E(K_\pi) = \{0\}$.  By Lemma~\ref{lem-gend}
(iii), $E$ is $\pi$-relatively generated by global sections.  So $E$
is $\pi$-relatively deformation ample.

\

\textbf{(iii)}: This is proved by induction on $n$, the case $n=1$
being tautological.  It suffices to consider the case when $T$ is
affine.  Suppose $n>1$ and suppose the result is known for $n-1$.  In
particular, $E^{\otimes(n-1)}$ is generated by global sections.  There
is a natural surjection
\begin{equation*}
\pi^*\pi_* \lt( E^{\otimes(n-1)} \rt)\otimes_{\OO_B} E \rightarrow
E^{\otimes n}. 
\end{equation*}
And there is a surjective map $\OO_T^{\oplus N}\rightarrow
\pi_*(E^{\otimes(n-1)})$.  Hence there is a surjection $E^{\oplus
N}\rightarrow E^{\otimes n}$.  By (ii) and induction, $E^{\oplus N}$
is $\pi$-relatively deformation ample.  By (i), the quotient
$E^{\otimes n}$ is $\pi$-relatively deformation ample.  Thus (iii) is
proved by induction.
\end{proof}

\begin{lem}~\label{lem-DAcrit}
  Let $B$ be a proper, connected, at-worst-nodal curve of arithmetic
  genus $0$ over an algebraically closed field $k$.  Let $E$ be a
  locally free sheaf of positive rank such that,
\begin{enumerate}

\item[(i)] 
for every irreducible component $B_i\subset B$, $E|_{B_i}$ is
generated by global sections, and

\item[(ii)] 
there exists a nonempty, connected, closed subcurve $B'\subset B$ such
that $E|_{B'}$ is deformation ample.

\end{enumerate}
Then $E$ is deformation ample.
\end{lem}

\begin{proof}
Let $\delta$ be the number of irreducible components of $B$ which are
not contained in $B'$.  The result is proved by induction on $\delta$.
The base case $\delta=0$ is tautological, for then $B=B'$.  Assume
that $\delta >0$ and that the result is true for all smaller values of
$\delta$.

\

Let $B_1\subset B$ be an irreducible component of $B$.  Let
$B_2\subset B$ denote the union of all irreducible components other
than $B_1$.  There exists an irreducible component $B_1$ not contained
in $B'$ such that $B_2$ is connected: if the dual graph of $B'$
contains every leaf (= vertex of valence 1) of the dual graph of $B$,
then the two graphs are equal.  The intersection $B_1 \cap B_2$ is a
single node, denoted $b$.  By the induction hypothesis, $E|_{B_2}$ is
deformation ample.

\

The claim is that $E$ is generated by global sections.  Denote by
$F\subset E$ the image of $H^0(B,E)\otimes_k \OO_B \rightarrow E$.
There is a short exact sequence of coherent sheaves:
\begin{equation*}
\begin{CD}
0 @>>> E\otimes_{\OO_B} \OO_{B_1}(-b) @>>> E @>>> E\otimes_{\OO_B}
\OO_{B_2} @>>> 0. 
\end{CD}
\end{equation*}
Since $E|_{B_1}$ is a locally free sheaf on $\PP^1$ generated by
global sections, Grothendieck's lemma and the cohomology of line
bundles on $\PP^1$ imply that $h^1(B,E\otimes_{\OO_B} \OO_{B_1}(-b)) =
0$ ($h^1(\PP^1,\OO_{\PP^1}(a))=0$ for $a \geq -1$).  Hence all the
global sections of $E|_{B_2}$ lift to global sections of $E$, i.e.,
$F\rightarrow E|_{B_2}$ is surjective.  So $E/F$ is supported on
$B_1$.  Hence $E/F$ is a quotient of $E|_{B_1}$.  Since $E|_{B_1}$ is
generated by global sections, also $E/F$ is generated by global
sections.  There is a short exact sequence
\begin{equation*}
\begin{CD}
0 @>>> F @>>> E @>>> E/F @>>> 0.
\end{CD}
\end{equation*}
By Lemma~\ref{lem-gend} (iii), $E$ is generated by global sections.

\

There is a short exact sequence of coherent sheaves,
\begin{equation*}
0 \rightarrow E(K_B)\otimes_{\OO_B} \OO_{B_2}(-b) \rightarrow E(K_B)
\rightarrow 
E(K_B)\otimes_{\OO_B} \OO_{B_1} \rightarrow 0.
\end{equation*}
This gives a long exact sequence in cohomology, part of which is,
\begin{equation*}
H^1(B,E(K_B) \otimes_{\OO_B} \OO_{B_2}(-b)) \rightarrow H^1(B,E(K_B))
\rightarrow H^1(B,E(K_B) \otimes_{\OO_B} \OO_{B_1}) 
\rightarrow 0.
\end{equation*}
The inclusion map $B_2 \rightarrow B$ is finite, therefore there is a
canonical isomorphism of $\OO_B$-modules (cf. ~\cite[Sec. III.6]{R+D},
~\cite[Cor. 5.68]{KM}),
\begin{equation*}
\OO_{B_2}(K_{B_2}) \cong
\textit{Hom}_{\OO_B}(\OO_{B_2},\OO_B(K_B)) \cong
\OO_B(K_B)\otimes_{\OO_B} \OO_{B_2}(-b).
\end{equation*}
Hence $h^1(B, E(K_B) \otimes_{\OO_B} \OO_{B_2}(-b) ) = h^1( B_2, E
\otimes_{\OO_B} \OO_{B_2}(K_{B_2}))$, which is zero by the induction
assumption.  Similarly, $E(K_B) \otimes_{\OO_B} \OO_{B_1}$ is
isomorphic to $E \otimes_{\OO_B} \OO_{B_1}(-1)$ (identifying $B_1$
with $\PP^1$).  Since $E|_{B_1}$ is generated by global sections, it
follows by Grothendieck's lemma and the cohomology of line bundles on
$\PP^1$ that $h^1(B_1,E|_{B_1}(-1)) =0$. Hence $h^1(B,E(K_B)) =0$, and
$E$ is deformation ample.  Therefore the lemma is proved by induction
on $\delta$.
\end{proof}

\begin{rmk} \label{rmk-DAcrit}
  A particular case of Lemma~\ref{lem-DAcrit} is when $B'$ is one
  irreducible component of $B$, in which case the lemma says that a
  locally free sheaf on $B$ which is \emph{generically ample} in the
  sense of Lazarsfeld ~\cite{F} is deformation ample.
\end{rmk}

\section{Deformation theory of stable maps} \label{sec-def}
The Kontsevich moduli space of genus $0$ stable maps,
$\Kgnb{0,r}{X,e}$, and the Behrend-Manin moduli spaces,
$\Kbm(X,\tau)$, are described in Subsection~\ref{subsec-kv}.  The
deformation theory of stable maps has been worked out in ~\cite{BF}
and ~\cite{B}.  Many specific deformation-theoretic results follow
easily from these papers and are known to the experts, but have not
been written down.  Some of these specific results are proved in this
section.  Although these results will only be applied to \emph{genus
$0$} stable maps in this paper, the same arguments work for stable
maps of arbitrary genus; in this section only, stable maps and stable
$A$-graphs are not necessarily assumed to be of genus $0$.

\

Let $X$ and $T$ be schemes.

\begin{defn} \label{defn-prestab} 
A \emph{family of $r$-pointed, genus $g$ prestable maps to $X$ over
  $B$} is a triple
\begin{equation*}
\zeta = ((p:\Sigma \rightarrow B,\sigma_1,\dots, \sigma_r), g:\Sigma
\rightarrow X)
\end{equation*}
consisting of a family $p:\Sigma \rightarrow B$ of prestable curves of
genus $g$ over $B$, a sequence of $r$ disjoint sections $\sigma_i:B
\rightarrow \Sigma$ with image contained in the smooth locus of $p$
(if $r=0$, the sections are omitted), and a morphism $g:\Sigma
\rightarrow X$.

Let $((\Sigma,\sigma_1,\dots,\sigma_r),g)$ be a prestable map over an
algebraically closed field.  An irreducible component $\Sigma_i
\subset \Sigma$ is \emph{stable} if the restriction of the
\emph{log-dualizing sheaf}, $\OO_{\Sigma}(K_{\Sigma}+\sigma_1 + \dots
+ \sigma_r)\otimes_{\OO_{\Sigma}} \OO_{\Sigma_i}$, is $g$-relatively
ample, i.e., one of the following hold,
\begin{enumerate}

\item[(i)] 
$g:\Sigma_i \rightarrow X$ is
nonconstant, 

\item[(ii)] 
$p_a(\Sigma_i) > 1$, 

\item[(iii)] 
$p_a(\Sigma_i) =1$ and $\Sigma_i$ contains at least one marked point
or external node of $\Sigma$, or

\item[(iv)]
$p_a(\Sigma_i)=0$ and $\Sigma_i$ contains at least three marked points
and nodes of $B$.

\end{enumerate}
The curve $\Sigma$ is \emph{stable} if
$\OO_{\Sigma}(K_{\Sigma}+\sigma_1+\dots+\sigma_r)$ is $g$-relatively
ample, i.e., every irreducible component of $\Sigma$ is stable.  The
family $\zeta$ is \emph{stable} if for each geometric point $t\in T$,
the curve $\Sigma_t$ is stable.
\end{defn}

Now assume that $X$ is smooth.

\begin{notat} \label{not-cpx}
Denote by $L_\zeta$ the complex of coherent sheaves on $\Sigma$
\begin{equation}
\begin{CD}
-1 & & 0 \\
g^*\Omega_X @> (dg)^\dagger >> \Omega_p(\sigma_1(T) + \dots + \sigma_r(T))
\end{CD}
\end{equation}

For a scheme $T$ and a bounded above complex of coherent sheaves $C$
on $T$, $C^\vee$ denotes the object in the derived category of
coherent sheaves on $T$,
\begin{equation*}
C^\vee := \mathbb{R}\textit{Hom}_{\OO_{T}}(C,\OO_T).
\end{equation*}
In particular, $L_\zeta^\vee$ is the object
\begin{equation*}
L_\zeta^\vee =
\mathbb{R}\textit{Hom}_{\OO_\Sigma}(L_\zeta,\OO_\Sigma)
\end{equation*} 
in the derived category of coherent sheaves on $\Sigma$.
\end{notat}

The relevance of the complex $L_\zeta^\vee$ is the following.

\begin{lem} \label{lem-cpx}
  Let $X$ be a smooth quasi-projective scheme.  Let
  $\Kgnb{g,r}{X,\beta}$ denote the Deligne-Mumford stack of
  $r$-pointed stable maps to $X$ of arithmetic genus $g$ and degree
  $\beta$.  Let $p:\Sigma \rightarrow \Kgnb{g,r}{X,\beta}$ denote the
  universal curve, let $\sigma_i: \Kgnb{g,r}{X,\beta} \rightarrow
  \Sigma$ denote the universal sections, and let $g:\Sigma \rightarrow
  X$ denote the universal map, i.e.
\begin{equation*}
\zeta = \lt( \lt( p:\Sigma \rightarrow
\Kgnb{g,r}{X,\beta}, 
\sigma_1, \dots, \sigma_r \rt), g:\Sigma \rightarrow X \rt)
\end{equation*}
is the universal family of stable maps.  There is an \emph{obstruction
  theory} for $\Kgnb{g,r}{X,\beta}$ in the sense of
  \cite[Def. 4.4]{BF} of the form
\begin{equation*}
 \lt( \mathbb{R}p_*(L_\zeta^\vee) [1] \rt)^\vee \rightarrow
L_{\Kgnb{g,r}{X,\beta}}.
\end{equation*}
A similar result holds for prestable maps, cf. Remark~\ref{rmk-cpx}.
\end{lem}

\begin{proof} Essentially this follows from ~\cite{BF} and
  ~\cite{B}.
\end{proof}

\begin{rmk} \label{rmk-cpx}
Explicitly, if $\zeta = ((\Sigma,\sigma_1,\dots,\sigma_r), g:\Sigma
\rightarrow X)$ is a stable map or a prestable map, the space of first
order deformations of $\zeta$ is $\EExt{\Sigma}^1(L_\zeta,\OO_\Sigma)$
and the obstruction group is a subgroup of
$\EExt{\Sigma}^2(L_\zeta,\OO_\Sigma)$.  In the case of a prestable
map, the space of infinitesimal automorphisms of the map is
$\EExt{\Sigma}^0(L_\zeta,\OO_\Sigma)$ (for stable maps this group is
zero).  In particular, if $\EExt{\Sigma}^2(L_\zeta,\OO_\Sigma)$
vanishes, then $\Kgnb{g,r}{X,\beta}$ is smooth at the point $[\zeta]$.
\end{rmk}

\begin{lem} \label{lem-stab}
Let $((\Sigma,\sigma_1,\dots,\sigma_r),g)$ be a prestable map.  Let
$\Theta_\Sigma$ denote the tangent sheaf of $\Sigma$, i.e. the dual of
$\Omega_\Sigma$.  The space of infinitesimal automorphisms of $\zeta$,
$\EExt{\Sigma}^0(L_\zeta,\OO_\Sigma)$, is canonically isomorphic to a
subspace of the space of infinitesimal automorphisms of
$(\Sigma,\sigma_1,\dots,\sigma_r)$,
$H^0(\Sigma,\Theta_\Sigma(-(\sigma_1 + \dots + \sigma_r)))$.

An irreducible component $\Sigma_i \subset \Sigma$ is stable iff the
restriction of every infinitesimal automorphism of $\zeta$ to
$\Theta_\Sigma(-(\sigma_1 + \dots + \sigma_r)) \otimes_{\OO_\Sigma}
\OO_{\Sigma_i}$ is zero.  Moreover, if $\Sigma_i$ is unstable, then
every infinitesimal automorphism of the nodal curve $\Sigma_i$ that
fixes all marked points and nodes of $\Sigma$ is the image of an
infinitesimal automorphism of $\zeta$.
\end{lem}

\begin{proof}

Analyzing the spectral sequence for hypercohomology,
$\EExt{\Sigma}^0(L_\zeta,\OO_\Sigma)$ is canonically isomorphic to the
kernel of
\begin{equation*}
\text{Ext}_{\OO_\Sigma}^0(\Omega_\Sigma(\sigma_1+\dots+\sigma_r),\OO_\Sigma)
\rightarrow 
\text{Ext}_{\OO_\Sigma}^0(g^*\Omega_X,\OO_\Sigma).
\end{equation*}
And $\text{Ext}_{\OO_\Sigma}^0 (\Omega_\Sigma (
\sigma_1+\dots+\sigma_r ), \OO_\Sigma )$ equals
$H^0(\Sigma,\Theta_\Sigma ( - (\sigma_1 + \dots + \sigma_r) ) )$.

\

Assume that $\Sigma_i$ is stable, i.e., $\Sigma_i$ satisfies one of
the Cases (i)--(iv) in Definition~\ref{defn-prestab}.  In Cases
(ii)--(iv), the image of
\begin{equation*}
H^0(\Sigma,\Theta_\Sigma(-(\sigma_1 + \dots + \sigma_r))) \rightarrow
H^0(\Sigma_i,\Theta_\Sigma(-(\sigma_1 + \dots + \sigma_r)) \otimes_{\OO_\Sigma}
\OO_{\Sigma_i} )
\end{equation*}
is zero, hence the restriction of every infinitesimal automorphism of
$\zeta$ is zero.  In Case (i), chasing through diagrams, the image of
$H^0(\Sigma,\Theta_\Sigma(-(\sigma_1 + \dots + \sigma_r)))$ is
contained in the subsheaf which is the kernel of $d(g|_{\Sigma_i}):
\Theta_{\Sigma_i} \rightarrow g^* T_X \otimes_{\OO_\Sigma}
\OO_{\Sigma_i}$.  Since $g$ is non-constant, this map is nonzero.  And
$\Theta_{\Sigma_i}$ is a torsion-free sheaf, so the kernel of
$d(g|_{\Sigma_i})$ is zero.  Hence, also in Case (i), the restriction
of every infinitesimal automorphism of $\zeta$ is zero.

\

Assume that $\Sigma_i$ is not stable.  Then $g$ contracts $\Sigma_i$
to a point, and either $\Sigma=\Sigma_i$ is a curve of arithmetic
genus $1$ and there are no marked points, or $\Sigma_i$ is a smooth
rational curve which contains at most two marked points and nodes of
$\Sigma$.  In both cases there is a positive dimensional group of
automorphisms of $\Sigma_i$ which fix all marked points and nodes of
$\Sigma$: $Pic^\tau(\Sigma_i)$ acting by translation if $\Sigma_i$ has
arithmetic genus $1$, and the group of automorphisms of $\PP^1$ fixing
(at most) two points if $\Sigma_i$ is a smooth rational curve.  The
Lie algebra of this positive dimensional group is the Lie algebra of
infinitesimal automorphisms of the marked curve $\Sigma_i$.  This
group of automorphisms of $\Sigma_i$ extends to a group of
automorphisms of the map $\zeta$.  Therefore every infinitesimal
automorphism of the marked curve $\Sigma_i$ extends to an
infinitesimal automorphism of $\zeta$.  In particular there exists an
infinitesimal automorphism of $\zeta$ whose restriction to $\Sigma_i$
is nonzero.
\end{proof}

Let $f:X \rightarrow Y$ be a smooth morphism of smooth
quasi-projective varieties.  Let $\zeta =
((\Sigma,\sigma_1,\dots,\sigma_r),g:\Sigma \rightarrow X)$ be a
prestable map.  Denote by $h:\Sigma \rightarrow Y$ the composition
$h=f\circ g$, and denote by $\xi$ the prestable map
$((\Sigma,\sigma_1,\dots,\sigma_r),h:\Sigma \rightarrow Y)$.  There is
a short exact sequence of complexes
\begin{equation*}
\begin{CD}
0 @>>> L_\xi @>>> L_\zeta @>>> g^* \Omega_f[1]
@>>> 0,
\end{CD}
\end{equation*}
defined by the following commutative diagram,
\begin{equation*}
\begin{CD}
 & & -1 & & 0 \\
L_\xi: & & g^* f^* \Omega_Y @> dh^\dagger >> \Omega_\Sigma(\sigma_1 +
\dots + \sigma_r) \\
@VVV @V g^*(df)^\dagger VV @VV = V \\
L_\zeta: & & g^* \Omega_X @> df^\dagger >> \Omega_\Sigma(\sigma_1 +
\dots + \sigma_r) \\
@VVV @VVV @VVV \\
g^*\Omega_f[1]: & & g^* \Omega_f @>>> 0 
\end{CD}
\end{equation*}

\begin{lem} \label{lem-XtoY}
\begin{enumerate}

\item[(i)] 
If $\xi$ is stable, then $\zeta$ is stable.

\item[(ii)]
If the dimension of the obstruction group of $\zeta$ is $0$, then the
dimension of the obstruction group of $\xi$ is $0$.

\item[(iii)]
If $h^1(\Sigma,g^*\Omega_f^\vee) = 0$ and if the dimension of the
obstruction group of $\xi$ is $0$, then the dimension of the
obstruction group of $\zeta$ is $0$, and the map from the space of
first order deformations of $\zeta$ to the space of first order
deformations of $\xi$ is surjective.

\end{enumerate}
\end{lem} 

\begin{proof}
\textbf{(i)}: If the log dualizing sheaf of
$(\Sigma,\sigma_1,\dots,\sigma_r)$ is $h$-ample, then it is $g$-ample.

\

\textbf{(ii)}: Associated to the short exact sequence of complexes
above, there is a long exact sequence of hyperExt, part of which is
\begin{equation*}
\begin{CD}
\EExt{\Sigma}^2(L_\zeta,\OO_{\Sigma}) @>>> \EExt{\Sigma}^2(L_\xi,
\OO_{\Sigma}) 
@>>> \EExt{\Sigma}^3(g*\Omega_f[1],\OO_{\Sigma}).
\end{CD}
\end{equation*}
Of course $\EExt{\Sigma}^k(g^*\Omega_f[1],\OO_\Sigma) =
H^{k-1}(\Sigma,g^*\Omega_f^\vee)$.  In particular $\text{dim}
\EExt{\Sigma}^3(g^* \Omega_f[1],\OO_\Sigma) =
h^2(\Sigma,g^*\Omega_f^\vee) = 0$.  Therefore if $\text{dim}
\EExt{\Sigma}^2(L_\zeta,\OO_\Sigma) = 0$, then $\text{dim}
\EExt{\Sigma}^2(L_\xi,\OO_\Sigma) = 0$.

\

\textbf{(iii)}: If $h^1(\Sigma,g^*\Omega_f^\vee) = 0$, then
$\EExt{\Sigma}^2(L_\zeta,\OO_\Sigma) \rightarrow
\EExt{\Sigma}^2(L_\xi,\OO_{\Sigma})$ is an isomorphism.  Hence if the
dimension of the obstruction group of $\zeta$ is $0$, then the
dimension of the obstruction group of $\xi$ is $0$.  Moreover the
preceding two terms in the long exact sequence of hyperExts give a
surjection $\EExt{\Sigma}^1(L_\zeta,\OO_{\Sigma}) \rightarrow
\EExt{\Sigma}^1(L_\xi, \OO_{\Sigma})$.
\end{proof}

Let $q:S \rightarrow B$ be a smooth morphism and let $i:\Sigma
\rightarrow S$ be an unramified morphism of $B$-schemes.  Then
$(g,i):\Sigma \rightarrow X \times S$ is an unramified morphism of
$B$-schemes, hence the map of coherent sheaves
\begin{equation*}
d(g,i)^\dagger:
(g,i)^* \Omega_(X\times S/B) \rightarrow \Omega_p 
\end{equation*}
is surjective.  Because $p$ is flat of relative dimension $1$ and the
geometric fibers are reduced, local complete intersection schemes, the
kernel of $d(g,i)^\dagger$ is a locally free sheaf $N^\vee_{(g,i)}$.
Similarly the kernel of $di^\dagger: i^* \Omega_q \rightarrow
\Omega_p$ is a locally free sheaf $N^\vee_{i}$.

\

There is a short exact sequence of coherent sheaves
\begin{equation*}
\begin{CD}
0 @>>> N^\vee_i @>>> N^\vee_{(g,i)} @>>> g^*\Omega_X @>>> 0.
\end{CD}
\end{equation*}
Denote by $N_{(\zeta,i)}^\vee$ the subsheaf of
$N^\vee_{(g,i)}(\sigma_1(B)+ \dots + \sigma_r(B))$ that contains
$N^\vee_i(\sigma_1(B)+\dots+\sigma_r(B))$ and such that
$N_{(\zeta,i)}^\vee/N^\vee_i(\sigma_1(B) + \dots +\sigma_r(B))$ is
identified with $g^*\Omega_X \subset g^*\Omega_X(\sigma_1(B) + \dots +
\sigma_r(B))$.  Observe that $N_{(\zeta,i)}^\vee$ is a locally free
sheaf.

\begin{notat} \label{notat-Lzetai}
There is a canonical map $N^\vee_{(g,i)}(\sigma_1(B)+ \dots +
\sigma_r(B)) \rightarrow (g,i)^* \Omega_{X\times S/B}$.  The two
projections give a canonical isomorphism of $\Omega_{X\times S/B}$
with $\pi_X^*\Omega_X \oplus \pi_S^* \Omega_q$.  Denote by
$\gamma_{(\zeta,i)}:N_{(\zeta,i)}^\vee \rightarrow
g^*\Omega_X(\sigma_1(B) + \dots + \sigma_r(B)) \oplus i^*
\Omega_q(\sigma_1(B) + \dots +\sigma_r(B))$ the induced morphism.
Observe that the composition of $\gamma_{(\zeta,i)}$ with projection
on the first summand factors through $g^* \Omega_X \subset g^*
\Omega_X(\sigma_1(B) + \dots + \sigma_r(B))$.  Denote by
$\alpha_{(\zeta,i)}:N^\vee_{(\zeta,i)} \rightarrow g^* \Omega_X$ and
$\beta_{(\zeta,i)}:N^\vee_{(\zeta,i)} \rightarrow i^*
\Omega_q(\sigma_1(B) + \dots + \sigma_r(B))$ the composition of
$\gamma_{(\zeta,i)}$ with the two projections. Denote by
$L_{(\zeta,i)}$ the complex of locally free $\OO_{\Sigma}$-modules
concentrated in degrees $[-1,0]$,
\begin{equation*}
\begin{CD}
-1 & & 0 \\
N_{(\zeta,i)}^\vee @> \beta_{(\zeta,i)} >> i^*
\Omega_q(\sigma_1(B)+\dots + \sigma_r(B)). 
\end{CD}
\end{equation*}
Denote by $\lambda_{(\zeta,i)}:L_{(\zeta,i)} \rightarrow L_\zeta$ the
quasi-isomorphism of complexes of coherent $\OO_{\Sigma}$-modules,
\begin{equation*}
\begin{CD}
N_{(\zeta,i)}^\vee @> \beta_{(\zeta,i)} >> i^* \Omega_q(\sigma_1(B)+
\dots + \sigma_r(B)) \\ @V \alpha_{(\zeta,i)} VV @VV (di)^\dagger V
. \\ g^* \Omega_X @> (dg)^\dagger >> \Omega_p(\sigma_1(B) + \dots +
\sigma_r(B))
\end{CD}
\end{equation*}
\end{notat}

The relevance of $\lambda_{(\zeta,i)}:L_{(\zeta,i)} \rightarrow
L_\zeta$ is that the complex $L^\vee_{(\zeta,i)}$ is easy to compute
since $L_{(\zeta,i)}$ is a complex of locally free sheaves; it is
simply
\begin{equation*}
\begin{CD}
0 & & 1 \\
i^* T_q(-(\sigma_1(B)+\dots + \sigma_r(B))) @> 
\beta_{(\zeta,i)}^\dagger >> N_{(\zeta,i)}.
\end{CD}
\end{equation*}
In most applications, $B$ will be the spectrum of a field and $S$ will
be a surface.

\subsection{Contracting unstable components}\label{subsec-unstable}
In this subsection, the base $B$ will always be the spectrum of an
algebraically closed field.  The changes necessary to get relative
versions of the lemmas over a more general base are straightforward.

\

Let
\begin{equation*}
\zeta = ((\Sigma,\sigma_1,\dots,\sigma_r),g:\Sigma \rightarrow X)
\end{equation*}
be a prestable map, let
\begin{equation*}
(\Sigma',\sigma'_1,\dots, \sigma'_r,\sigma''_1,\dots,\sigma''_s)
\end{equation*}
be a proper, connected, at-worst-nodal curve, and let
\begin{equation*}
u:(\Sigma',\sigma'_1,\dots, \sigma'_r) \rightarrow
(\Sigma,\sigma_1,\dots, \sigma_r)
\end{equation*} 
be a map which contracts some of the unstable components of
$(\Sigma',\sigma'_1,\dots,\sigma'_r)$ (i.e., $u$ contracts some of the
irreducible components of $\Sigma'$ which have arithmetic genus $0$
and contain fewer than three nodes and marked points).  Denote $g' =
g\circ u$ and denote by $\zeta'$ the prestable map
\begin{equation*}
\zeta' = ((\Sigma',\sigma'_1,\dots, \sigma'_r, \sigma''_1, \dots,
\sigma''_s), g':\Sigma' \rightarrow 
X).
\end{equation*}
What is the relationship of $\EExt{\Sigma}(L_\zeta,\OO_\Sigma)$ and
$\EExt{\Sigma'}(L_{\zeta'}, \OO_{\Sigma'})$?

\

Any morphism $u:\Sigma'\rightarrow \Sigma$ as above can be factored as
a sequence of \emph{elementary morphisms}, defined below.

\begin{defn} \label{defn-elemtypes}
\begin{enumerate}

\item[(i)]
The morphism $u$ is a \emph{Type I elementary morphism} if $\zeta =
(\Sigma,g:\Sigma\rightarrow X)$ is a prestable map without marked
points, $u:\Sigma' \rightarrow \Sigma$ contracts a single unstable
component to a smooth point of $\Sigma$, and $\zeta' = (\Sigma', g' =
g\circ u:\Sigma' \rightarrow X)$.

\item[(ii)]
The morphism $u$ is a \emph{Type II elementary morphism} if $\zeta =
(\Sigma,g:\Sigma \rightarrow X)$ is a prestable map without marked
points, $u:\Sigma' \rightarrow \Sigma$ contracts a single unstable
component to a node of $\Sigma$, and $\zeta' = (\Sigma', g' = g\circ
u:\Sigma' \rightarrow X)$.

\item[(iii)]
The morphism $u$ is a \emph{Type III elementary morphism} if $\zeta =
(\Sigma,(\sigma_1,\dots,\sigma_r),g:\Sigma \rightarrow X)$ is a marked
prestable map, $\zeta'$ is the same prestable map but with one extra
marked point, and $u:\Sigma'\rightarrow \Sigma$ is the identity map.

\end{enumerate}
\end{defn}

Let $S$ be a smooth surface and let $i:\Sigma \rightarrow S$ be a
closed immersion.  Let $s\in \Sigma \subset S$ be a closed point.

\begin{notat} \label{notat-S}
Denote by $v:S'\rightarrow S$ the blowing up of $S$ at $s$.  Denote by
$E \subset S$ the exceptional divisor. Denote by $i':\Sigma'
\rightarrow S'$ the \emph{reduced total transform of $B$}; i.e., the
reduced scheme of $v^{-1}(\Sigma)$.  Denote by $v$ also the morphism
of pairs $v:(S',\Sigma') \rightarrow (S,\Sigma)$.  Denote by
$u:\Sigma' \rightarrow \Sigma$ the restriction of $v$ to $\Sigma'$.
Denote by $g':\Sigma' \rightarrow X$ the composition $g' = g \circ u$.
Denote by $\zeta'$ the prestable map $(\Sigma',g':\Sigma' \rightarrow
X)$.  Denote by $\Gamma \subset \Sigma'$ the closed (not necessarily
connected) subcurve which is the union of all irreducible components
other than $E$, and denote $D = E\cap \Gamma$.
\end{notat}

\begin{defn} \label{defn-moretypes}
\begin{enumerate}

\item[(i)]
The morphism of pairs is \emph{Type Ia}
if $s\in \Sigma$ is a smooth point that lies on a stable component.

\item[(ii)]
The morphism of pairs is \emph{Type Ib}
if $s\in \Sigma$ is a smooth point that lies on an unstable component.

\item[(iii)]
The morphism of pairs is \emph{Type I} if it is Type Ia or Type Ib.

\item[(iv)]
The morphism of pairs is \emph{Type IIa} if $s\in \Sigma$ is a node,
and there exists a first order deformation of $\zeta$ that smoothes the
node $s$ (to first order).

\item[(v)]
The morphism of pairs is \emph{Type IIb} if $s\in \Sigma$ is a node,
and there is no first order deformation of $\zeta$ that smoothes the
node $s$ (to first order).

\item[(vi)]
The morphism of pairs is \emph{Type II} if it is Type IIa or Type IIb.

\end{enumerate}
\end{defn}

If $v$ is Type I, then $v^* \Sigma = \Sigma'$ as Cartier divisors.  If
$v$ is Type II, then $v^* \Sigma = \Sigma' + E$ as Cartier divisors.


\begin{lem} \label{lem-hypercohom}
For every integer $k$ the pullback morphism of sheaf hypercohomology
groups, $\mathbb{H}^k(\Sigma,L_\zeta^\vee) \rightarrow
\mathbb{H}^k(\Sigma',\mathbb{L}u^* L_\zeta^\vee)$, is an isomorphism.
\end{lem}

\begin{proof}
Of course $v_*\OO_{S'} = \OO_S$ and $R^{k}v_* \OO_{S'} = \{0\}$ if
$k>0$.  Also $v_* (\OO_{S'}(E)) = \OO_S$ and $R^{k}v_* (\OO_{S'}(E)) =
\{0\}$ for $k>0$.  If $v$ is Type I then $\OO_{S'}(-\Sigma') = v^*
\OO_S(-\Sigma)$, and if $v$ is Type II then $\OO_{S'}(-\Sigma') =
v^*\OO_S(-\Sigma) \otimes_{\OO_{S'}} \OO_{S'}(E)$.  For both types,
the projection formula implies that $v_*( \OO_{S'}(-\Sigma') ) =
\OO_S(-\Sigma)$ and $R^{k}v_*(\OO_{S'}(-\Sigma')) = \{0\}$ for $k>0$.

\
  
Associated to the short exact sequence of coherent $\OO_{S'}$-modules,
\begin{equation*}
\begin{CD}
0 @>>> \OO_{S'}(-\Sigma') @>>> \OO_{S'} @>>> \OO_{\Sigma'} @>>> 0,
\end{CD}
\end{equation*}
there is a long exact sequence of higher direct images $R^k v_*$, and
the higher direct images of $\OO_{S'}$ and $\OO_{S'}(-\Sigma')$ have
just been computed.  The conclusion is that $u_* \OO_{\Sigma'} =
\OO_\Sigma$ and $R^{k}u_* \OO_{\Sigma'} = \{ 0 \}$ for $k>0$.  In
other words, the canonical morphism of complexes of coherent
$\OO_\Sigma$-modules, $\OO_\Sigma[0] \rightarrow \mathbb{R}u_*
\OO_{\Sigma'}$, is a quasi-isomorphism.  From this and the projection
formula, it follows that the canonical morphism
\begin{equation*}
L_{(\zeta,i)}^\vee \rightarrow 
\mathbb{R}u_* \mathbb{L}u^* (L_{(\zeta,i)}^\vee)
\end{equation*}
is a quasi-isomorphism.  Therefore the pullback morphisms
\begin{equation*}
\mathbb{H}^k(\Sigma,L_\zeta^\vee) \rightarrow
\mathbb{H}^k(\Sigma',\mathbb{L}u^*L_\zeta^\vee)
\end{equation*}
are isomorphisms.
\end{proof}

There is a canonical map of coherent sheaves $\OO_{S'}(\Sigma')
\rightarrow v^* \OO_S(\Sigma)$.  It is an isomorphism if $v$ is Type
I, and is injective with cokernel $v^*\OO_S(\Sigma) \otimes_{\OO_{S'}}
\OO_E$ if $v$ is Type II.  If $v$ is Type II, then $v^*\OO_X(\Sigma)
\otimes_{\OO_{S'}} \OO_E \cong M \otimes_\CC \OO_E$ where $M =
\OO_S(\Sigma)|_s$ is a one-dimensional vector space.  If $v$ is Type
I, the canonical morphism $N_{(\zeta',i')} \rightarrow u^*
N_{(\zeta,i)}$ is an isomorphism.  If $v$ is Type II, there is an
exact sequence:
\begin{equation*}
0 \rightarrow M \otimes_\CC \textit{Tor}_1^{\OO_{S'}}(\OO_{\Sigma'},\OO_E)
\rightarrow N_{(\zeta',i')} \rightarrow u^* N_{(\zeta,i)} \rightarrow M
\otimes_\CC \OO_E \rightarrow 0
\end{equation*}


\begin{lem} \label{lem-torcomp}
Let $N_{E/S'}^\vee$ denote the conormal sheaf of $E \subset S'$.
There is a canonical isomorphism
$\textit{Tor}_1^{\OO_{S'}}(\OO_{\Sigma'}, \OO_E) \cong N_{E/S'}^\vee
\otimes_{\OO_E} \OO_E(-D)$.
\end{lem}

\begin{proof}
There is an $\OO_{S'}$-flat resolution of $\OO_E$,
\begin{equation*}
\begin{CD}
0 @>>> \OO_{S'}(-E) @>>> \OO_S @>>> \OO_E @>>> 0.
\end{CD}
\end{equation*}
Tensoring this resolution with $\OO_{\Sigma'}$ over $\OO_{S'}$ gives a
canonical isomorphism,
\begin{equation*}
\textit{Tor}_1^{\OO_{S'}}(\OO_{\Sigma'},\OO_E) \cong {\mc
  I}_{\Gamma}\otimes_{\OO_{S'}}\OO_{S'}(-E) \cong N_{E/S'}^\vee
  \otimes_{\OO_E} \OO_E(-D).
\end{equation*}
where ${\mc I}_{\Gamma} \subset \OO_{\Sigma'}$ is the ideal sheaf of 
$\Gamma \subset \Sigma'$.
\end{proof}

In particular, if $v$ is Type II there is an exact sequence:
\begin{equation*}
0 \rightarrow M \otimes_\CC N_{E/S'}^\vee \otimes_{\OO_E} \OO_E(-D)
\rightarrow N_{(\zeta',i')} 
\rightarrow u^*N_{(\zeta,i)} \rightarrow M \otimes_\CC \OO_E \rightarrow
0.
\end{equation*}

\begin{lem} \label{lem-anotherles}
There is a long exact sequence,
\begin{equation*}
0 \rightarrow T_E(-D) \rightarrow (i')^*T_{S'} \xrightarrow{dv} u^*
i^* T_S \rightarrow T_E \otimes_{\OO_E} N_{E/S'} \rightarrow 0.
\end{equation*}
(Both if $v$ is Type I and if $v$ is Type II.)
\end{lem}

\begin{proof}
For both types, there is a short exact sequence of coherent
$\OO_{S'}$-modules
\begin{equation*}
\begin{CD}
0 @>>> v^* \Omega_S @> (dv)^\dagger >> \Omega_{S'} @>>> \Omega_E @>>>
0.
\end{CD}
\end{equation*}
There is an associated long exact sequence of higher derived functors
of $\textit{Hom}_{\OO_{S'}}(\cdot,\OO_{S'})$, part of which is the
short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> T_{S'} @>>> v^* T_S @>>>
\textit{Ext}^1_{\OO_{S'}}(\Omega_E,\OO_{S'}) @>>> 0. 
\end{CD}
\end{equation*}
The resolution of $\OO_E$ from the proof of Lemma~\ref{lem-torcomp}
gives a canonical isomorphism,
\begin{equation*}
\textit{Ext}^1_{\OO_{S'}}(\OO_E,\OO_{S'}) \cong N_{E/S'},
\end{equation*}
where $N_{E/S'}$ is the dual of $N_{E/S'}^\vee$.  So the previous
exact sequence is,
\begin{equation*}
\begin{CD}
0 @>>> T_{S'} @>>> v^* T_S @>>> T_E\otimes_{\OO_E} N_{E/S'} @>>> 0
\end{CD}
\end{equation*}
Of course $T_E \otimes_{\OO_E} N_{E/S'}$ is a locally free
$\OO_E$-module; in particular, we have a canonical isomorphism,
\begin{equation*}
\textit{Tor}_1^{\OO_{S'}}(\OO_{\Sigma'},T_E
\otimes_{\OO_E} N_{E/S'}) \cong
\textit{Tor}_1^{\OO_{S'}}(\OO_{\Sigma'},\OO_E) \otimes_{\OO_E} T_E
\otimes_{\OO_E} N_{E/S'}.
\end{equation*}
Tensoring the short exact sequence above with $\OO_{\Sigma'}$ over
$\OO_{S'}$ and using Lemma~\ref{lem-torcomp}, produces the exact
sequence,
\begin{equation*}
0 \rightarrow T_E(-D) \rightarrow (i')^*T_{S'} \xrightarrow{dv} u^*
i^* T_S \rightarrow T_E\otimes_{\OO_E} N_{E/S'} \rightarrow 0.
\end{equation*}
\end{proof}

\

The maps $N_{(\zeta',i')} \rightarrow u^* N_{(\zeta,i)}$ and
$(i')^*T_{S'} \rightarrow u^* i^* T_S$ are compatible with
$\alpha_{(\zeta',i')}^\dagger$ and $u^*\alpha_{(\zeta,i)}^\dagger$.
So there is an induced map of complexes $L_{(\zeta',i')}^\vee
\rightarrow u^* L_{(\zeta,i)}^\vee$.

\begin{notat} \label{notat-cpxs}
\begin{enumerate}

\item[(i)]
Denote by $du:L_{(\zeta',i')}^\vee \rightarrow h^* L_{(\zeta,i)}^\vee$
the induced map of complexes.

\item[(ii)]
Denote by $\text{Image}(du) \hookrightarrow u^* L_{(\zeta,i)}^\vee$
the image of $du$ in the Abelian category of complexes of coherent
$\OO_{\Sigma'}$-modules.

\item[(iii)]
If $v$ is Type I, denote $K_I = T_E(-D)[0]$ and $Q_I = \lt( T_E
\otimes_{\OO_E} N_{E/S'} \rt)[0]$.

\item[(iv)]
If $v$ is Type II, denote
\begin{equation*}
\begin{array}{llc}
K_{II} & = & T_E(-D)[0] \oplus\lt( M \otimes_{\CC} N_{E/S'}^\vee(-D)
\rt) [-1] \\
Q_{II} & = & \lt( T_E \otimes_{\OO_E} N_{E/S'} \rt)[0] \oplus 
\lt(M \otimes_\CC \OO_E \rt)[-1]
\end{array} 
\end{equation*}

\end{enumerate}
\end{notat}

\begin{lem} \label{lem-sescpx}
Both if $v$ is Type I and if $v$ is Type II, there are short exact
sequences of complexes of coherent $\OO_{\Sigma'}$-modules,
\begin{equation*}
\begin{CD}
0 @>>> K @>>> L_{(\zeta',i')}^\vee @> du >> \text{Image}(du) @>>> 0 \\
0 @>>> \text{Image}(du) @>>> u^* L_{\zeta}^\vee @>>> Q @>>> 0
\end{CD}
\end{equation*}
\end{lem}

\begin{proof}
If $v$ is Type I, there is a commutative diagram with exact rows,
\begin{equation*}
\begin{array}{ccccccccccc}
 & & & & L_{(\zeta',i')}^\vee & \xrightarrow{du}& u^* L_{(\zeta,i)}^\vee \\
\\
0 & \rightarrow & T_E(-D) & \rightarrow & (i')^*T_{S'} &
\xrightarrow{(i')^*dv} & u^*i^* T_S & \rightarrow & T_E
\otimes_{\OO_E} N_{E/S'} & \rightarrow & 0 \\  
 & & & & \downarrow & & \downarrow & \\
 & & & & N_{(\zeta',i')} & \cong & h^* N_{(\zeta,i)}
\end{array}
\end{equation*}
The middle two columns of this diagram give $du:L_{(\zeta',i')}^\vee
\xrightarrow{} u^* L_{(\zeta,i)}^\vee$.  By inspection the kernel of
$du$ is $K_I$ and the cokernel of $du$ is $Q_I$.

\

If $v$ is Type II, there is a commutative diagram with exact rows,
\begin{equation*}
\begin{array}{ccccccccccc}
 & & & & L_{(\zeta',i')}^\vee & \xrightarrow{du} & u^* L_{(\zeta,i)}^\vee \\
\\ 
0 & \rightarrow & T_E(-D) & \rightarrow & (i')^*T_{S'} &
\xrightarrow{(i')^*dv} & u^*i^* T_S & \rightarrow & T_E
\otimes_{\OO_E} N_{E/S'} & \rightarrow & 0 \\  
 & & & & \downarrow & &  \downarrow & \\
0 & \rightarrow &  M\otimes_\CC N_{E/S'}^\vee(-D) & \rightarrow &
N_{(\zeta',i')} & \longrightarrow & u^* N_{(\zeta,i)} & 
\rightarrow & M\otimes_\CC \OO_E  & \rightarrow & 0
\end{array}
\end{equation*}
The middle two columns of this diagram give $du:L_{(\zeta',i')}^\vee
\rightarrow u^* L_{(\zeta,i)}^\vee$.

\

The claim is that the induced map of kernels, $T_E(-D) \rightarrow
M\otimes_\CC N_{E/S'}^\vee(-D)$, is zero, and the induced map of
cokernels, $T_E\otimes_{\OO_E} N_{E/S'} \rightarrow M\otimes_\CC
\OO_E$ is zero.  To see this, observe that in each map, the domain and
target are locally free sheaves on $E$.  So each map is really a
section of a $\textit{Hom}_{\OO_E}(\cdot,\cdot)$ sheaf.  Up to
canonical isomorphisms, the sheaf in both cases is simply
\begin{equation*}
\textit{Hom}_{\OO_E}(T_E,M\otimes_\CC N_{E/S'}^\vee) \cong
M\otimes_\CC \Omega^2_{S'}\otimes_{\OO_S} \OO_E.
\end{equation*}
Identifying $E$ with $\PP^1$, there are isomorphisms $N_{E/S'} \cong
\OO_E(-1)$ and $\OO_E(K_E) \cong \OO_E(-2)$.  By the adjunction
formula for divisors on surfaces $\Omega^2_{S'}\otimes_{\OO_S} \OO_E =
\OO_{S'}(K_{S'}) \otimes_{\OO_S} \OO_E$ is isomorphic to $\OO_E(K_E)
\otimes_{\OO_E} N_{E/S'}^\vee$, i.e., $\Omega^2_{S'}\otimes_{\OO_S}
\OO_E \cong \OO_E(-2+1) = \OO_E(-1)$.  Since this sheaf has no nonzero
global sections, both the induced maps are zero.  Therefore the kernel
of $du$ is $K_{II}$ and cokernel of $du$ is $Q_{II}$.
\end{proof}

Assume first that $v$ is Type I.  Since $T_E(-D) \cong \OO_E(1)$,
$\mathbb{H}^0(\Sigma',K_I) = H^0(E,T_E(-D))$ is $2$-dimensional, and
$\text{dim} \mathbb{H}^{i}(\Sigma',K_I)= 0$ for $i\neq 0$.  Similarly
$\mathbb{H}^0(\Sigma',Q_I) = H^0(E,T_E(E))$ is $2$-dimensional and
$\text{dim} \mathbb{H}^{i}(\Sigma',Q_I)= 0$ for $i>0$.  Therefore
there is a long exact sequence of hypercohomology groups:
\begin{eqnarray*}
0 \rightarrow H^0(E,T_E(-D)) \rightarrow
\mathbb{H}^0(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow 
\mathbb{H}^0(\Sigma, L_{(\zeta,i)}^\vee) \rightarrow \dots \\ 
\dots  \rightarrow  H^0(E, T_E\otimes_{\OO_E} N_{E/S'}) \rightarrow
\mathbb{H}^1(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow
 \mathbb{H}^1(\Sigma, L_{(\zeta,i)}^\vee ) \rightarrow 0 \\
  0 \rightarrow \mathbb{H}^2(\Sigma', L_{(\zeta',i')}^\vee) 
\rightarrow \mathbb{H}^2(\Sigma, L_{(\zeta,i)}^\vee ) \rightarrow 0
\end{eqnarray*}

\begin{lem} \label{lem-def1a}
If $v$ is Type Ia then there are exact sequences,
\begin{eqnarray*}
0 \rightarrow H^0(E,T_E(-D)) \rightarrow
\mathbb{H}^0(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow 
\mathbb{H}^0(\Sigma, L_{(\zeta,i)}^\vee) \rightarrow 0, \\ 
0 \rightarrow  H^0(E, T_E\otimes_{\OO_E} N_{E/S'}) \rightarrow
\mathbb{H}^1(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow
 \mathbb{H}^1(\Sigma, L_{(\zeta,i)}^\vee ) \rightarrow 0, \\
  0 \rightarrow \mathbb{H}^2(\Sigma',L_{(\zeta',i')}^\vee) 
\rightarrow \mathbb{H}^2(\Sigma, L_{(\zeta,i)}^\vee ) \rightarrow 0.
\end{eqnarray*}
In other words, the canonical map from the space of infinitesimal
automorphisms of $\zeta'$ to the space of infinitesimal automorphisms
of $\zeta$ is surjective with $2$-dimensional kernel, the canonical
map from the space of first order deformations of $\zeta'$ to the
space of first order deformations of $\zeta$ is surjective with
$2$-dimensional kernel, and the obstruction space of $\zeta'$ equals
the obstruction space of $\zeta$.
\end{lem}

\begin{proof}
The only claim that doesn't follow from the long exact sequence of
cohomology is that $\mathbb{H}^0(\Sigma',L_{(\zeta',i')}^\vee)
\rightarrow \mathbb{H}^0(\Sigma, L_{(\zeta,i)}^\vee)$ is surjective.
Since the irreducible component $\Sigma_j \subset \Sigma$ containing
$s$ is stable, Lemma~\ref{lem-stab} states that every infinitesimal
automorphism of $\zeta$ vanishes on $\Sigma_j$.  The infinitesimal
automorphisms of $\zeta$ that vanish at $s$ are the same as the
infinitesimal automorphisms of $\zeta'$ that vanish on $E$.  Therefore
every infinitesimal automorphism of $\zeta$ is the image of an
infinitesimal automorphism of $\zeta'$ that vanishes on $E$.
\end{proof}

Let $v$ be a morphism of Type Ib.  Let $\Sigma_j \subset \Sigma$
denote the unstable component containing $s$.  Let $N$ denote the
$1$-dimensional vector space $\Theta_{\Sigma_j}
\otimes_{\OO_{\Sigma_j}} \OO_s$.

\begin{lem} \label{lem-def1b}
If $v$ is Type Ib then there are exact sequences,
\begin{eqnarray*}
0 \rightarrow H^0(E,T_E(-D)) \rightarrow
\mathbb{H}^0(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow 
\mathbb{H}^0(\Sigma, L_{(\zeta,i)}^\vee) \rightarrow N \rightarrow 0 \\ 
0 \rightarrow  H^0(E, T_E \otimes_{\OO_E} N_{E/S'})/N \rightarrow
\mathbb{H}^1(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow
 \mathbb{H}^1(\Sigma, L_{(\zeta,i)}^\vee ) \rightarrow 0 \\
  0 \rightarrow \mathbb{H}^2(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow 
\mathbb{H}^2(\Sigma, L_{(\zeta,i)}^\vee ) \rightarrow 0
\end{eqnarray*}
In other words, the canonical map from the space of infinitesimal
automorphisms of $\zeta'$ to the space of infinitesimal automorphisms
of $\zeta$ has a $2$-dimensional kernel and a $1$-dimensional
cokernel, the canonical map from the space of first order deformations
of $\zeta'$ to the space of first order deformations of $\zeta$ is
surjective with $1$-dimensional kernel, and the obstruction space of
$\zeta'$ equals the obstruction space of $\zeta$.
\end{lem}

\begin{proof}
As in Lemma~\ref{lem-def1a}, the only claim that doesn't follow from
the long exact sequence of cohomology is that the cokernel of
$du:\mathbb{H}^0(\Sigma', L_{(\zeta',i')}^\vee) \rightarrow
\mathbb{H}^0(\Sigma,L_{(\zeta,i)}^\vee)$ is $N$.  Let $\Sigma'_j
\subset \Sigma'$ denote the strict transform of $\Sigma_j$;
$\Sigma'_j$ is canonically isomorphic to $\Sigma_j$.  Composing $du$
with the restriction map $\mathbb{H}^0(\Sigma,L_{(\zeta,i)}^\vee)
\rightarrow H^0(\Sigma_j,\Theta_{\Sigma_j})$ gives a map which is
canonically isomorphic to the restriction map
$\mathbb{H}^0(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow
H^0(\Sigma'_j,\Theta_{\Sigma'_j})$.  By Lemma~\ref{lem-stab}, every
infinitesimal automorphism of $\zeta'$ restricts to an infinitesimal
automorphism of $\Sigma'_j$ that vanishes at $s$.  On the other hand,
the infinitesimal automorphisms of $\zeta$ that vanish at $s$ are the
same as the infinitesimal automorphisms of $\zeta'$ that vanish on
$E$.  Hence the image of $du$ is precisely the subspace of
infinitesimal automorphisms of $\zeta$ that vanish at $s$.

\


Since $\Sigma_j$ is unstable, either $\Sigma = \Sigma_j$ is a curve of
arithmetic genus $1$ with no markings and $g$ is constant, or
$\Sigma_j$ is a smooth curve of genus $0$ containing at most $2$
marked points and nodes of $\Sigma$ and $g|_{\Sigma_j}$ is constant.
In each case, it is easy to see that there is an infinitesimal
automorphism of $\Sigma_j$ that vanishes at all marked points and
nodes of $\Sigma$ and that does not vanish at $s$.  By
Lemma~\ref{lem-stab}, this infinitesimal automorphism of $\Sigma_j$ is
the image of an infinitesimal automorphism of $\zeta$, i.e., there
exists an infinitesimal automorphism of $\zeta$ that does not vanish
at $s$.  Therefore the cokernel of $du$ is $N$.
\end{proof}

Assume next that $v$ is Type II.  Because the divisor $D \subset E$
has degree $2$, the $\OO_E$-module $N_{E/S'}^\vee(-D)$ is isomorphic
to $\OO_E(-1)$.  Since $h^0(E,\OO_E(-1)) = h^1(E,\OO_E(-1)) = 0$, the
term $M\otimes_\CC N_{E/S'}^\vee(-D)[-1]$ in $K_{II}$ does not
contribute to the hypercohomology; i.e., the hypercohomology of
$K_{II}$ is the sheaf cohomology of $T_E(-D)$.  The $\OO_E$-module
$T_E(-D)$ is isomorphic to $\OO_E$; hence $h^0(E,T_E(-D))=1$ and
$h^1(E,T_E(-D))=0$.  Therefore $\text{dim}
\mathbb{H}^0(\Sigma',K_{II}) = H^0(E,T_E(-D))$ is $1$-dimensional, and
$\text{dim} \mathbb{H}^{i}(\Sigma',K_{II}) =0 $ for $k\neq 1$.

\

For $Q_{II}$ both $T_E\otimes_{\OO_E} N_{E/S'}[0]$ and $M\otimes_\CC
\OO_E[-1]$ contribute to the hypercohomology.  The $\OO_E$-module
$T_E\otimes_{\OO_E} N_{E/S'}$ is isomorphic to $\OO_E(1)$.  Hence
$\mathbb{H}^0(\Sigma',Q_{II}) = H^0(E,T_E(E))$ is $2$-dimensional,
$\mathbb{H}^1(\Sigma',Q_{II}) = M$ is $1$-dimensional (recall $M =
\OO_S(\Sigma)|_s$), and $\text{dim}\mathbb{H}^k(\Sigma',Q_{II}) =0$
for $k\neq 0,1$.  Therefore there is a long exact sequence in
hypercohomology:
\begin{eqnarray*}
0 \rightarrow H^0(E,T_E(-D)) \rightarrow
\mathbb{H}^0(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow
\mathbb{H}^0(\Sigma,L_{(\zeta,i)}^\vee) \rightarrow \dots \\
\dots \rightarrow H^0(E,T_E\otimes_{\OO_E} N_{E/S'}) \rightarrow
\mathbb{H}^1(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow
\mathbb{H}^1(\Sigma,L_{(\zeta,i)}^\vee) \rightarrow \dots \\
\dots \rightarrow M \rightarrow \mathbb{H}^2(\Sigma',L_{(\zeta',i')}^\vee)
\rightarrow \mathbb{H}^2(\Sigma,L_{(\zeta,i)}^\vee) \rightarrow 0
\end{eqnarray*}

\

Every infinitesimal automorphism of $\zeta$ vanishes at $s$.  So the
infinitesimal automorphisms of $\zeta$ are the same as the
infinitesimal automorphisms of $\zeta'$ that vanish on $E$.  In
particular, $\mathbb{H}^0(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow
\mathbb{H}^0(\Sigma,L_{(\zeta,i)}^\vee)$ is surjective.  The map
$\mathbb{H}^1(\Sigma,L_{(\zeta,i)}^\vee) \rightarrow \OO_S(\Sigma)|_s$
is nonzero iff there are deformations of $\zeta$ that smooth the node
$s$ to first order.  This proves the following two lemmas, which are
stated separately for notational convenience.

\begin{lem} \label{lem-def2a}
If $v$ is Type IIa then there are exact sequences,
\begin{eqnarray}
0 \rightarrow H^0(E,T_E(-D)) \rightarrow
\mathbb{H}^0(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow
\mathbb{H}^0(\Sigma,L_{(\zeta,i)}^\vee) \rightarrow 0 \\
0 \rightarrow H^0(E,T_E \otimes_{\OO_E} N_{E/S'}) \rightarrow
\mathbb{H}^1(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow
\mathbb{H}^1(\Sigma,L_{(\zeta,i)}^\vee) \rightarrow 
\OO_S(\Sigma)|_s \rightarrow 0 \\
0 \rightarrow
\mathbb{H}^2(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow 
\mathbb{H}^2(\Sigma, L_{(\zeta,i)}^\vee) \rightarrow 0 
\end{eqnarray}
In other words, the canonical map from the space of infinitesimal
automorphisms of $\zeta'$ to the space of infinitesimal automorphisms
of $\zeta$ is surjective with a $1$-dimensional kernel, the canonical
map from the space of first order deformations of $\zeta'$ to the
space of first order deformations of $\zeta$ has a $2$-dimensional
kernel and a $1$-dimensional cokernel, and the obstruction space to
$\zeta'$ equals the obstruction space to $\zeta$.
\end{lem}

\begin{lem} \label{lem-def2b}
If $v$ is Type IIb then there are exact sequences,
\begin{eqnarray}
0 \rightarrow H^0(E,T_E(-D)) \rightarrow
\mathbb{H}^0(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow
\mathbb{H}^0(\Sigma,L_{(\zeta,i)}^\vee) \rightarrow 0 \\
0 \rightarrow H^0(E,T_E \otimes_{\OO_E} N_{E/S'}) \rightarrow
\mathbb{H}^1(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow
\mathbb{H}^1(\Sigma,L_{(\zeta,i)}^\vee) \rightarrow 0 \\
0 \rightarrow \OO_S(\Sigma)|_s \rightarrow
\mathbb{H}^2(\Sigma',L_{(\zeta',i')}^\vee) \rightarrow 
\mathbb{H}^2(\Sigma,
L_{(\zeta,i)}^\vee) \rightarrow 0. 
\end{eqnarray}
In other words, the canonical map from the space of infinitesimal
automorphisms of $\zeta'$ to the space of infinitesimal automorphisms
of $\zeta$ is surjective with a $1$-dimensional kernel, the canonical
map from the space of first order deformations of $\zeta'$ to the
space of first order deformations of $\zeta$ is surjective with
$1$-dimensional kernel, and the map from the obstruction space of
$\zeta'$ to the obstruction space of $\zeta$ is surjective and has a
$1$-dimensional kernel.
\end{lem}

\

Finally, assume that $u:\Sigma'\rightarrow \Sigma$ is a Type III
elementary morphism, i.e., $u$ is the identity map, but there is one
marked point $\sigma'\in \Sigma'$ that is not in $\Sigma$.

\begin{defn} \label{defn-yetmoretypes}
Let $u:\Sigma' \rightarrow \Sigma$ be a Type III elementary morphism.
\begin{enumerate}

\item[(i)]
The morphism $u$ is \emph{Type IIIa} if $\sigma' \in \Sigma$ lies on
an unstable component.

\item[(ii)]
The morphism $u$ is \emph{Type IIIb} if $\sigma' \in \Sigma$ lies on a
stable component.

\end{enumerate}
\end{defn} 
  
In both cases, there is a canonical short exact sequence of complexes:
\begin{equation*}
\begin{CD}
0 @>>> L_\zeta @>>> L_{\zeta'} @>>> 
\Omega_\Sigma(\sigma') \otimes_{\OO_\Sigma} \OO_{\sigma'}[0] @>>> 0
\end{CD}
\end{equation*}
Of course $\Ext_{\OO_\Sigma}^1(\Omega_\Sigma(\sigma')
\otimes_{\OO_\Sigma} \OO_{\sigma'},\OO_\Sigma)$ is canonically
isomorphic to $T_\Sigma \otimes_{\OO_\Sigma} \OO_{\sigma'}$, and
$\text{dim}\Ext_{\OO_\Sigma}^k(\Omega_\Sigma(\sigma')
\otimes_{\OO_\Sigma} \OO_{\sigma'}, \OO_\Sigma) = 0$ for $k \neq 1$.
In particular, there is an induced map
$$\Ext^0_{\OO_\Sigma}(L_\zeta,\OO_\Sigma)
\rightarrow T_\Sigma \otimes_{\OO_\Sigma} \OO_{\sigma'}.$$ 
This map is zero iff $\sigma'$ lies on a stable component
of $\zeta$.  Combined with the long exact sequence of hypercohomology
associated to the short exact sequences, this proves the following two
lemmas, which are stated separately for notational convenience.

\begin{lem} \label{lem-def3a}
If $u:\Sigma'\rightarrow \Sigma$ is Type IIIa then there are exact
  sequences,
\begin{eqnarray*}
0 \rightarrow \EExt{\Sigma'}^0(L_{\zeta'},\OO_{\Sigma'}) \rightarrow
\EExt{\Sigma}^0(L_\zeta,\OO_\Sigma) \rightarrow T_\Sigma
\otimes_{\OO_\Sigma} 
\OO_{\sigma'} \rightarrow 0 \\
0 \rightarrow \EExt{\Sigma'}^1(L_{\zeta'},\OO_{\Sigma'}) \rightarrow
\EExt{\Sigma}^1(L_\zeta,\OO_\Sigma) \rightarrow 0 \\
0 \rightarrow \EExt{\Sigma'}^2(L_{\zeta'},\OO_{\Sigma'}) \rightarrow
\EExt{\Sigma}^2(L_\zeta,\OO_\Sigma) \rightarrow 0 
\end{eqnarray*}
In other words, the space of infinitesimal automorphisms of $\zeta'$
maps isomorphically to a codimension $1$ linear subspace of the space
of infinitesimal automorphisms of $\zeta$, the space of first order
deformations of $\zeta'$ equals the space of first order deformations
of $\zeta$, and the obstruction space of $\zeta'$ equals the
obstruction space of $\zeta$.
\end{lem}

\begin{lem} \label{lem-def3b}
If $u:\Sigma'\rightarrow \Sigma$ is Type IIIb then there are exact
  sequences,
\begin{eqnarray*}
0 \rightarrow \EExt{\Sigma'}^0(L_{\zeta'},\OO_{\Sigma'}) \rightarrow
\EExt{\Sigma}^0(L_\zeta,\OO_\Sigma) \rightarrow 0 \\
0 \rightarrow T_\Sigma \otimes_{\OO_\Sigma} \OO_{\sigma'} 
\rightarrow \EExt{\Sigma'}^1(L_{\zeta'},\OO_{\Sigma'}) 
\rightarrow \EExt{\Sigma}^1(L_\zeta,\OO_\Sigma) \rightarrow 0 \\
0 \rightarrow \EExt{\Sigma'}^2(L_{\zeta'},\OO_{\Sigma'}) \rightarrow
\EExt{\Sigma}^2(L_\zeta,\OO_\Sigma) \rightarrow 0 
\end{eqnarray*}
In other words, the space of infinitesimal automorphisms of $\zeta'$
equals the space of infinitesimal automorphisms of $\zeta$, the
canonical map from the space of first order deformations of $\zeta'$
to the space of first order deformations of $\zeta$ is surjective with
$1$-dimensional kernel, and the obstruction space of $\zeta'$ equals
the obstruction space of $\zeta'$.
\end{lem}

Taken together, Lemma~\ref{lem-def1a} through Lemma~\ref{lem-def3b},
describe the canonical maps
$\EExt{\Sigma'}^k(L_{\zeta'},\OO_{\Sigma'}) \rightarrow
\EExt{\Sigma}^k(L_\zeta,\OO_\Sigma)$ for any morphism
$u:\Sigma'\rightarrow \Sigma$ that removes a subset of marked points
from $\Sigma'$ and then contracts a subset of the unstable components.

\subsection{Gluing stable maps}\label{subsec-glue}
Let $\tau$ be a stable $A$-graph, not necessarily of genus $0$.  To
describe the analogue of Lemma~\ref{lem-cpx}, a more precise
description of the stack $\Kbm(X,\tau)$ is needed.  Let $B$ be a
scheme.  A $1$-morphism $\widetilde{\zeta}:B \rightarrow \Kbm(X,\tau)$
is equivalent to a pair $(\zeta,\phi)$ consisting of a family of
$r$-pointed stable maps,
\begin{equation*}
\zeta = ((p:\Sigma \rightarrow B,\sigma_1,\dots, \sigma_r), g:\Sigma
\rightarrow X),
\end{equation*}
along with a natural assignment to each geometric point $b\in B$ of a
contraction of stable $A$-graphs $\phi_b:\tau(\zeta_b) \rightarrow
\tau$.  The graph $\tau(\zeta_b)$ is the stable $A$-graph of
$((\Sigma_b,\sigma_1(b),\dots,\sigma_r(b)), g_b)$.  A
\emph{contraction} of stable $A$-graphs, $\phi:\tau'\rightarrow \tau$
is a map that contracts subgraphs of $\tau'$ to vertices of $\tau$,
cf. ~\cite[Def. 1.8]{BM}.  Geometrically the vertices of $\tau$ give a
decomposition of $\Sigma_b$ into connected subcurves.  The main
example of a contraction of stable $A$-graphs comes from a family of
stable maps over a discrete valuation ring, $\zeta:\SP R \rightarrow
\Kgnb{g,r}{X,e}$. Let $\zeta_0$ denote the fiber over the geometric
closed point of $R$ and let $\zeta_\eta$ denote the fiber over the
geometric generic point of $R$.  There is a canonical contraction
$\phi_{(\zeta,R)}:\tau(\zeta_0) \rightarrow \tau(\zeta_\eta)$; a
vertex of $\tau(\zeta_0)$ corresponding to an irreducible component
$\Sigma_{0,j} \subset \Sigma_0$ maps to a vertex of $\tau(\zeta_\eta)$
corresponding to an irreducible component of $\Sigma_{\eta,k} \subset
\Sigma_\eta$ iff $\Sigma_{0,j}$ is in the closure of
$\Sigma_{\eta,k}$.  The assignment $\phi$ from above is called
\emph{natural} if it is compatible with the action of the Galois group
of $\kappa(b)$ and for each map from a discrete valuation ring to $B$,
$\rho: \SP R \rightarrow B$, the contractions $\phi_{\rho(0)}$ and
$\phi_{\rho(\eta)}$ commute with the contraction
$\phi_{\rho^*\zeta,R}$.

\begin{notat} \label{notat-Neps}
Associated to each edge $\epsilon=\{f_1,f_2\}$ of $\tau$, there is a
section $\sigma_\epsilon:B \rightarrow \Sigma$ such that for each
geometric point $b\in B$, $\sigma_\epsilon(b) \in \Sigma_b$ is a node.
Denote by $N_\epsilon$ the pullback $\sigma_\epsilon^*
\textit{Ext}^1_{\OO_\Sigma}(\Omega_p(\sigma_1(B)+\dots + \sigma_r(B)),
\OO_\Sigma)$.  Denote by $N_\tau$ the direct sum $\oplus_{\epsilon}
N_\epsilon$ where $\epsilon$ ranges over all edges of $\tau$.
\end{notat}

\begin{defn} \label{defn-gtrunc}
Let $T$ be a scheme.  Let $C$ be a complex of coherent sheaves on $T$
and let $n$ be an integer.  The \emph{good $(\leq r)$-truncation},
$C^{\leq r}$, is the complex of coherent sheaves on $T$,
\begin{equation*}
(C^{\leq r})^k = \lt\{ \begin{array}{ll}
                          C^k & k < r, \\
                          \text{Ker}(d^r:C^r \rightarrow C^{r+1}) &
                          k=r, \\
                          \{0\} & k > r
                       \end{array} \rt.
\end{equation*}
The differentials on $C^{\leq r}$ are the obvious ones.  The
association $C \mapsto C^{\leq r}$ defines a functor on the category
of complexes of coherent sheaves on $T$.  This functor takes
quasi-isomorphisms to quasi-isomorphisms and sends null-homotopic maps
to null-homotopic maps, thus it induces a well-defined functor on the
derived category of coherent sheaves on $T$.  There is a natural
transformation to the identity functor, $C^{\leq r} \rightarrow C$.
\end{defn}

The $\OO_B$-module $N_\epsilon$ is invertible and the localization of
$\textit{Ext}^1_{\OO_\Sigma}(\Omega_p(\sigma_1(B) + \dots +
\sigma_r(B)), \OO_\Sigma)$ along $\sigma_\epsilon(B)$ is canonically
isomorphic to $(\sigma_\epsilon)_* N_\epsilon$.  By construction there
is a map of complexes of coherent $\OO_\Sigma$-modules, $L_\zeta^\vee
\rightarrow \mathbb{R}\textit{Hom}_{\OO_\Sigma}( \Omega_p(\sigma_1(B)+
\dots + \sigma_r(B)), \OO_\Sigma)$.  For all $k>1$,
$\textit{Ext}^k_{\OO_B}(\Omega_B(\sigma_1(B) + \dots + \sigma_r(B)),
\OO_B) = \{0\}$.  So the complex is quasi-isomorphic to its $(\leq
1)$-good truncation,
\begin{equation*}
\mathbb{R}^{\leq 1}\textit{Hom}_{\OO_\Sigma}( \Omega_p(\sigma_1(B) +
\dots + \sigma_r(B)), \OO_\Sigma)
\xrightarrow{\text{qism}}
\mathbb{R}\textit{Hom}_{\OO_\Sigma}( \Omega_p(\sigma_1(B) + \dots +
\sigma_r(B)), \OO_\Sigma).
\end{equation*}

\begin{notat} \label{notat-abcde}
After replacing $L_\zeta^\vee$ by a quasi-isomorphic complex, there is
a morphism of complexes denoted $a_{(\zeta,\tau)}:L_\zeta^\vee
\rightarrow \mathbb{R}^{\leq 1}\textit{Hom}_{\OO_\Sigma}(
\Omega_p(\sigma_1(B)+ \dots + \sigma_r(B)) ,\OO_\Sigma)$ factoring the
original morphism.  There is a map of complexes of
$\OO_\Sigma$-modules denoted
\begin{equation*}
b_{(\zeta,\tau)}:
\mathbb{R}^{\leq 1}\textit{Hom}_{\OO_\Sigma}( \Omega_p(\sigma_1(B) +
\dots + \sigma_r(B)), \OO_\Sigma) \rightarrow
\textit{Ext}^1_{\OO_\Sigma}(\Omega_p(\sigma_1(B) + \dots +
\sigma_r(B)),
\OO_\Sigma)[-1].
\end{equation*}
Denote by, 
\begin{equation*}
c_{(\zeta,\tau)}: \textit{Ext}^1_{\OO_\Sigma}(
\Omega_p(\sigma_1(B)+\dots + \sigma_r(B), 
\OO_\Sigma)_\tau
\rightarrow \textit{Ext}^1_{\OO_\Sigma}( \Omega_p(\sigma_1(B) + \dots
+ \sigma_r(B)), \OO_\Sigma),
\end{equation*}
the kernel of,
\begin{equation*}
\textit{Ext}^1_{\OO_\Sigma}( \Omega_p(\sigma_1(B) + \dots +
\sigma_r(B)),
\OO_\Sigma) \rightarrow
\oplus_{\epsilon} (\sigma_\epsilon)_* N_\epsilon.
\end{equation*}
Denote by
\begin{equation*}
d_{(\zeta,\tau)}:\mathbb{R}^{\leq 1}\textit{Hom}_{\OO_\Sigma}(
\Omega_p(\sigma_1(B)+ \dots + \sigma_r(B)), \OO_\Sigma)_\tau \rightarrow
\mathbb{R}^{\leq 1}\textit{Hom}_{\OO_\Sigma}( \Omega_p(\sigma_1(B) +
\dots + \sigma_r(B)), \OO_\Sigma),
\end{equation*}
the fiber product of $b_{(\zeta,\tau)}$ and $c_{(\zeta,\tau)}$.  Denote by,
\begin{equation*}
e_{(\zeta,\tau)}:L_{(\zeta,\tau)}^\vee \rightarrow L_\zeta^\vee,
\end{equation*}
the fiber product of $a_{(\zeta,\tau)}$ and $d_{(\zeta,\tau)}$.  
\end{notat}

Let $|\tau|$ denote the underlying \emph{modular graph} of $\tau$,
i.e., the graph of $\tau$ along with the genus function on vertices,
but without the degree functon on vertices, cf. ~\cite{BM}.  There is
a (highly non-separated) Artin stack of $|\tau|$-prestable curves,
$\mathfrak{M}(|\tau|)$, and a $1$-morphism $\Kbm(X,\tau) \rightarrow
\mathfrak{M}(|\tau|)$.  The \emph{relative obstruction theory} for
this $1$-morphism is described in ~\cite{B}.  From this the absolute
obstruction theory of $\Kbm(X,\tau)$ readily follows.

\begin{lem} \label{lem-taucpx}
Let $X$ be a smooth quasi-projective scheme.  Let $\tau$ be a stable
$A$-graph.  Let
\begin{equation*}
\zeta = ((p:\Sigma \rightarrow \Kbm(X,\tau),
\sigma_1,\dots, \sigma_r), g:\Sigma \rightarrow X)
\end{equation*}
denote the universal family of stable maps over $\Kbm(X,\tau)$.  There
exists an \emph{obstruction theory} for $\Kbm(X,\tau)$ in the sense of
~\cite[Def. 4.4]{BF} of the form
\begin{equation*}
\lt( \mathbb{R}p_*(L_{(\zeta,\tau)}^\vee)[1] \rt)^\vee
\rightarrow L_{\Kbm(X,\tau)}.
\end{equation*}
This obstruction theory is perfect, and there is a distinguished
triangle in the derived category of complexes of coherent
$\OO_{\Kbm(X,\tau)}$-modules,
\begin{equation*}
\begin{CD}
\mathbb{R}p_*(L_{(\zeta,\tau)}^\vee)[1] 
@>>> \mathbb{R}p_*(L_\zeta^\vee)[1] @>>>
N_\tau @>>> \mathbb{R}p_*(L_{\zeta,\tau}^\vee)[2].
\end{CD}
\end{equation*}
\end{lem}

\begin{rmk} \label{rmk-simpler}
The definition of $L_{(\zeta,\tau)}^\vee$ seems very complicated, but
in fact it is quite simple.  Let
$((\Sigma,\sigma_1,\dots,\sigma_r),g)$ be a stable $\tau$-map.  Let
$i:\Sigma \rightarrow S$ be a closed immersion from $\Sigma$ to a
smooth surface $S$.  Let $N_i$ denote the normal sheaf of $i$.  The
complex $L_\zeta^\vee$ is quasi-isomorphic to the complex
$L_{(\zeta,i)}^\vee$.  There is a surjective map $N_{(\zeta,i)}
\rightarrow N_i(\sigma_1(B) + \dots + \sigma_r(B))$.  Define
$N_{(\zeta,i,\tau)} \subset N_{(\zeta,i)}$ to be the subsheaf which is
the inverse image of $\mc{I}\cdot N_i(\sigma_1(B) + \dots +
\sigma_r(B)) \subset N_i(\sigma_1(B) + \dots + \sigma_r(B))$, where
$\mc{I}$ is the ideal sheaf of $\cup_{\epsilon} \sigma_\epsilon$.
Then $e_{(\zeta,\tau)}:L_{(\zeta,\tau)}^\vee \rightarrow L_\zeta^\vee$
is quasi-isomorphic to the following map of complexes.
\begin{equation*}
\begin{CD}
 & & 0 & & 1 \\
L_{(\zeta,i,\tau)}^\vee: & & i^* T_S(\sigma_1(B)+ \dots + \sigma_r(B))
@>>> N_{(\zeta,i,\tau)} \\
@V e_{(\zeta,i,\tau)} VV  @V = VV @VVV \\
L_{(\zeta,i)}^\vee: & & i^* T_S(\sigma_1(B) + \dots + \sigma_r(B)) 
@>>> N_{(\zeta,i)}
\end{CD} 
\end{equation*}
\end{rmk}

Let $\epsilon = \{f_1,f_2\}$ is a disconnecting edge of $\tau$.  Let
$\tau_1\subset \tau$ be the maximal connected subgraph which contains
$f_1$ and not $f_2$, and let $\tau_2\subset \tau$ be the maximal
connected subgraph which contains $f_2$ and not $f_1$.  There are
``forgetful'' $1$-morphisms $F_i:\Kbm(X,\tau) \rightarrow
\Kbm(X,\tau_i)$ for $i=1,2$.  Let $\widetilde{\zeta}=(\zeta,\phi)$ be
a map in $\Kbm(X,\tau_i)$, where $\zeta =
((\Sigma,\sigma_1,\dots,\sigma_r),g)$.  Let $\sigma_\epsilon \in
\Sigma$ be the node corresponding to $\epsilon$.  Let
$\widetilde{\zeta}_i = (\zeta_i,\phi_i)\in \Kbm(X,\tau_i)$ be the
image $F_i(\widetilde{\zeta})$.  Denote $\zeta_i =
((\Sigma_i,\sigma_{i,0},\dots,\sigma_{i,r_i}),g_i)$ for $i=1,2$ where
$\sigma_{i,0} \in \Sigma_i$ is the point corresponding to the flag
$f_i$ of $\tau_i$, i.e., $\sigma_{i,0} = \sigma_\epsilon$.

\begin{lem}\label{lem-glue1}
  If
\begin{enumerate} 

\item[(i)] 
the dimension of the obstruction group of $\Kbm(X,\tau_i)$ at
$\widetilde{\zeta_i}$ is $0$, $i=1,2$, and

\item[(ii)]
the evaluation morphism $\text{ev}_{f_1}:\Kbm(X,\tau_1) \rightarrow X$
is smooth at $\widetilde{\zeta}_1$,

\end{enumerate}
then the dimension of the obstruction group of $\Kbm(X,\tau)$ at
$\widetilde{\zeta}$ is $0$, and there is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> \zeta_1^*T_{\text{ev}_{f_1}} @>>> \zeta^* T_{\Kbm(X,\tau)} @>>>
\zeta_2^* T_{\Kbm(X,\tau_2)} @>>> 0,
\end{CD}
\end{equation*}
where $T_{\text{ev}_{f_1}}$ is the dual of the sheaf of relative
differentials of $\text{ev}_{f_1}$.
\end{lem}

\begin{proof}
The proof follows from the fact that $\Kbm(X,\tau)$ is an open
substack of the $2$-fibered product:
\begin{equation*}
\Kbm(X,\tau_1) \times_{\text{ev}_{f_1},X,\text{ev}_{f_2}}
\Kbm(X,\tau_2).
\end{equation*}
\end{proof}

Let $\phi:\tau \rightarrow \tau'$ be the minimal contraction of stable
$A$-graphs that contracts the edge $\{f_1,f_2\}$.  The induced
$1$-morphism,
\begin{equation*}
\Kbm(X,\phi):\Kbm(X,\tau) \rightarrow \Kbm(X,\tau'),
\end{equation*}
is unramified and the image has codimension at most $1$.  In some
circumstances, this morphism is the normalization of a Cartier
divisor.

\begin{lem}\label{lem-glue2}
If
\begin{enumerate}

\item[(i)] 
$\tau$ has genus $0$,

\item[(ii)] 
the dimension of the obstruction group of $\Kbm(X,\tau_2)$ at
$\widetilde{\zeta}_2$ is $0$, and

\item[(iii)]
the $\OO_{\Sigma_1}$-module $g_1^* T_X$ is generated by global
sections,

\end{enumerate}
then the dimension of the obstruction group of $\Kbm(X,\tau')$ at
$\Kbm(X,\phi)(\widetilde{\zeta})$ is $0$, the irreducible component of
$\Kbm(X,\tau)$ containing $\widetilde{\zeta}$ maps to a Cartier
divisor under $\Kbm(X,\phi)$, and there is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> T_{\Kbm(X,\tau)}|_{\widetilde{\zeta}}
@>>> T_{\Kbm(X,\tau')}|_{\widetilde{\zeta}}
@>>>
\Theta_{\Sigma_1}|_{\sigma_{1,0}} \otimes 
\Theta_{\Sigma_2}|_{\sigma_{2,0}}
@>>> 0.
\end{CD}
\end{equation*}
\end{lem}

\begin{proof}
There is a short exact sequence of complexes,
\begin{equation*}
\begin{CD}
0 @>>> L_{\Sigma}(\sigma_1+\dots+\sigma_r) @>>> L_\zeta @>>>
g^*\Omega_X[1] @>>> 0, 
\end{CD}
\end{equation*}
that gives rise to a long exact sequence in hypercohomology, part of
which is,
\begin{equation*}
\mathbb{H}^1(\Sigma,L_\zeta^\vee) \rightarrow
\mathbb{H}^1(\Sigma,L_\Sigma^\vee(-(\sigma_1+\dots+\sigma_r))
\rightarrow H^1(\Sigma,g^*T_X) \rightarrow \mathbb{H}^2(\Sigma,L_\zeta^\vee)
\rightarrow 0.
\end{equation*}
The goal is to prove that $\text{dim}\mathbb{H}^2(\Sigma,L_\zeta^\vee)
= 0$, i.e., that $\mathbb{H}^1( \Sigma, L_\Sigma^\vee( -
(\sigma_1+\dots+\sigma_r ) ) ) \rightarrow H^1(\Sigma,g^*T_X)$ is
surjective.  After replacing $L_\Sigma$ and $L_\zeta$ by
quasi-isomorphic complexes of locally free $\OO_\Sigma$-modules, there
is a diagram of distinguished triangles of coherent $\OO_B$-modules,
\begin{equation*}
\begin{array}{l}
\begin{array}{ccccc}
L_\Sigma(\sigma_1+\dots+\sigma_r) & \rightarrow & L_\zeta &
\rightarrow & \dots \\ 
\downarrow & & \downarrow & & \\
L_\Sigma(\sigma_1+\dots+\sigma_r) \otimes_{\OO_\Sigma}
\OO_{\Sigma_2}  & \rightarrow & L_\zeta \otimes_{\OO_\Sigma}
\OO_{\Sigma_2} & \rightarrow &  \dots \\
\end{array} \\
\\
\\
\begin{array}{ccccc}
\dots & \rightarrow & g^* \Omega_X[1] & \rightarrow &
L_\Sigma(\sigma_1+\dots+\sigma_r) [-1] \\
& & \downarrow & & \downarrow \\
\dots & \rightarrow & g^* \Omega_X \otimes_{\OO_\Sigma}
\OO_{\Sigma_2}[1] & \rightarrow & 
L_\Sigma (\sigma_1+\dots+\sigma_r)
\otimes_{\OO_\Sigma} \OO_{\Sigma_2}[-1]
\end{array},
\end{array}
\end{equation*}
that gives rise to a commutative diagram of exact sequences in
hypercohomology,
\begin{equation*}
\begin{array}{ccc}
\mathbb{H}^1(\Sigma,L_\Sigma^\vee ( - (\sigma_1+\dots+\sigma_r))
) & \rightarrow &
\mathbb{H}^1(\Sigma,g^*T_X) \\
\downarrow & & \downarrow  \\
\mathbb{H}^1(\Sigma,L_\Sigma^\vee (-(\sigma_1+\dots+\sigma_r))  
\otimes_{\OO_\Sigma} \OO_{\Sigma_2}) & \rightarrow &
\mathbb{H}^1(\Sigma, g^*T_X \otimes_{\OO_\Sigma} \OO_{\Sigma_2})
\end{array} .
\end{equation*}
To prove the top horizontal arrow is surjective, it suffices to prove 
\begin{enumerate}

\item[(i)] 
the left vertical arrow is surjective,

\item[(ii)] 
the right vertical arrow is an isomorphism, and

\item[(iii)] 
the bottom vertical arrow is surjective.

\end{enumerate}

\

Let $S$ be a smooth surface, and let $i:\Sigma \rightarrow S$ be an
unramified morphism.  Then $L_\Sigma^\vee(
-(\sigma_1+\dots+\sigma_r))$ is represented in the derived category of
coherent sheaves by the complex
\begin{equation*}
\begin{CD}
& 0 & & 1 \\
L_i^\vee(-(\sigma_1+\dots+\sigma_r)): & i^*T_S( -
(\sigma_1+\dots+\sigma_r))  @> \beta_i^\dagger >> N_i ( -
(\sigma_1+\dots+\sigma_r) ). 
\end{CD}
\end{equation*}
Let $K$ denote the kernel of $\beta_i^\dagger$ and let $Q$ denote the
cokernel of $\beta_i^\dagger$.  The sheaf $K$ is torsion-free and is
locally free of rank $1$ on a dense open subset of $\Sigma$.  The
sheaf $Q$ is torsion.  There is a short exact sequence of complexes
\begin{equation*}
\begin{CD}
0 @>>> K[0] @>>> L_i^\vee(- (\sigma_1+\dots+\sigma_r) ) @>>>
Q[-1] @>>> 0.
\end{CD}
\end{equation*}
This gives rise to a long exact sequence in hypercohomology, part of
which is
\begin{equation*}
\begin{CD}
H^1(\Sigma, K) @>>> \mathbb{H}^1(\Sigma, L_i^\vee (
-(\sigma_1+\dots+\sigma_r) ) @>>> H^0(\Sigma, Q) @>>> 0.
\end{CD}
\end{equation*}
Let $K'$ denote the kernel of $\beta_i^\dagger \otimes_{\OO_\Sigma}
\OO_{\Sigma_2}$ and let $Q'$ denote the cokernel.  There are induced
maps of coherent sheaves on $\Sigma$, $K \rightarrow K'$ and $Q
\rightarrow Q'$.  These maps give rise to a commutative diagram with
exact rows,
\begin{equation*}
\begin{CD}
H^1(\Sigma, K) @>>> \mathbb{H}^1(\Sigma, L_i^\vee (
-(\sigma_1+\dots+\sigma_r) ) @>>> H^0(\Sigma, Q) @>>> 0 \\
@VVV @VVV @VVV \\
H^1(\Sigma, K') @>>> \mathbb{H}^1(\Sigma, L_i^\vee (
-(\sigma_1+\dots+\sigma_r) \otimes_{\OO_\Sigma} \OO_{\Sigma_2} )
@>>> 
H^0(\Sigma, Q') @>>> 0.
\end{CD}
\end{equation*}
Now $Q \rightarrow Q'$ is a surjective map of torsion sheaves, so the
map on the right is surjective.  And the cokernel of $K \rightarrow
K'$ is torsion, and hence the map on the left is surjective.
Therefore the map in the middle is surjective, i.e., (i) holds.

\

There is a short exact sequence of $\OO_\Sigma$-modules,
\begin{equation*}
\begin{CD}
0 @>>> (g_1)^* T_X(-\sigma_{1,0}) @>>> g^* T_X @>>> g^* T_X
\otimes_{\OO_\Sigma} \OO_{\Sigma_2} @>>> 0
\end{CD}
\end{equation*}
By hypothesis $(g_1)^* T_X$ is generated by global sections, hence
$h^1(\Sigma_1,(g_1)^* T_X(-\sigma_{1,0})) = 0$ by Lemma~\ref{lem-gend}.
Therefore $H^1(\Sigma,g^* T_X) \rightarrow H^1(\Sigma,g^* T_X
\otimes_{\OO_\Sigma}  
\OO_{\Sigma_2})$ is an isomorphism, i.e., (ii) is true.

\

Denote by $j:\Sigma_2 \rightarrow \Sigma$ the canonical closed
immersion.  There is a morphism of complexes of $\OO_\Sigma$-modules,
\begin{equation*}
\begin{CD}
& & -1 & & 0 \\
L_\zeta: & & g^* \Omega_X @> (dg)^\dagger >> \Omega_\Sigma(\sigma_1 +
\dots + \sigma_r) \\
@VVV @VVV @VV (dj)^\dagger V \\
j_* L_{\zeta_2}:  & & j_*(g_2)^* \Omega_X @> j_*(dg_2)^\dagger >> j_*
\Omega_\Sigma(\sigma_{2,0} + \dots + \sigma_{2,r_2}) 
\end{CD}
\end{equation*}
By adjunction, there is a morphism of complexes of
$\OO_{\Sigma_2}$-modules, $L_\zeta \otimes_{\OO_\Sigma} \OO_{\Sigma_2}
\rightarrow L_{\zeta_2}$.  So there is a commutative diagram in
hypercohomology
\begin{equation*}
\begin{CD}
\mathbb{H}^1(\Sigma_2,L_{\Sigma_2}^\vee( - (\sigma_1 + \dots +
\sigma_r) ) @>>> H^1(\Sigma_2,(g_2)^* T_X) \\
@VVV @VVV \\
\mathbb{H}^1(\Sigma_2,L_\Sigma^\vee ( - (\sigma_{2,0} + \dots +
\sigma_{2,r_2} ) )\otimes_{\OO_\Sigma} \OO_{\Sigma_2}) @>>>
H^1(\Sigma_2,j^*g^* T_X)
\end{CD}
\end{equation*}
Of course $(g_2)^* T_X = j^* g^* T_X$, so $H^1(\Sigma_2, (g_2)^* T_X)
\rightarrow H^1(\Sigma_2,j^* g^* T_X)$ is an isomorphism.  By
hypothesis, the obstruction group of $\zeta_2$ vanishes, hence
$\mathbb{H}^1(\Sigma_2,L_{\Sigma_2}^\vee ( - (\sigma_{2,0} + \dots +
\sigma_{2,r_2})) ) \rightarrow H^1(\Sigma_2,(g_2)^* T_X)$ is
surjective.  Therefore also $\mathbb{H}^1(\Sigma_2,L_\Sigma^\vee ( -
(\sigma_1 + \dots + \sigma_r)) \otimes_{\OO_\Sigma} \OO_{\Sigma_2})
\rightarrow H^1(\Sigma_2,j^* g^* T_X)$ is surjective.  This proves
(iii).  Therefore the dimension of the obstruction group of
$\Kbm(X,\tau')$ at $\Kbm(X,\phi)(\widetilde{\zeta})$ is $0$.

\

Because the obstruction groups of $\Kbm(X,\tau)$ and $\Kbm(X,\tau')$
both vanish, each stack is smooth of the expected dimension at
$\widetilde{\zeta}$.  The expected dimension of $\Kbm(X,\tau)$ is $1$
less than the expected dimension of $\Kbm(X,\tau')$, because $\tau$
has one extra edge (before stabilization).  Therefore the image of the
irreducible component of $\Kbm(X,\tau)$ containing $\widetilde{\zeta}$
is a Cartier divisor in $\Kbm(X,\tau')$.

\

Finally, the short exact sequence follows from the $2$-Cartesian
diagram of Artin stacks,
\begin{equation*}
\begin{CD}
\Kbm(X,\tau) @> \Kbm(X,\phi) >> \Kbm(X,\tau') \\
@VVV @VVV \\
\mathfrak{M}(|\tau|) @> \mathfrak{M}(|\phi|) >>
\mathfrak{M}(|\tau'|).
\end{CD}
\end{equation*}
The image of $\mathfrak{M}(|\phi|)$ is a Cartier divisor.  Because the
image of $\Kbm(X,\phi)$ is a Cartier divisor, the normal bundle of
$\Kbm(X,\phi)$ is simply the pullback of the normal bundle of
$\mathfrak{M}(|\phi|)$, which is the bundle whose fiber at
$\widetilde{\zeta}$ is the space of first-order deformations of the
node $\sigma_\epsilon \in \Sigma$.  It is well-known that this space
is canonically isomorphic to $\Theta_{\Sigma_1}|_{\sigma_{1,0}}
\otimes \Theta_{\Sigma_2}|_{\sigma_{2,0}}$.
\end{proof}

\begin{rmk} \label{rmk-glue2}
There is a relative version of Lemma~\ref{lem-glue2}.  Let $B$ be a
scheme and let $\widetilde{\zeta}=(\zeta,\phi)$ be a family of
$\tau$-maps over $B$.  If for every geometric point $b\in B$, the map
$\widetilde{\zeta}_b$ satisfies the hypotheses of
Lemma~\ref{lem-glue2}, then the obstruction group of $\Kbm(X,\sigma)$
at $\widetilde{\zeta}_b$ vanishes, the image of $\Kbm(X,\tau)$ in
$\Kbm(X,\tau')$ is a Cartier divisor, and there exists a short exact
sequence of locally free $\OO_B$-modules,
\begin{equation*}
\begin{CD}
0 @>>> \widetilde{\zeta}^* T_{\Kbm(X,\tau)} @>>> \widetilde{\zeta}^*
T_{\Kbm(X,\tau')} @>>> \sigma_{1,0}^* \Theta_{\Sigma_1/B} \otimes_{\OO_B}
\sigma_{2,0}^* \Theta_{\Sigma_2/B} @>>> 0.
\end{CD}
\end{equation*}
\end{rmk}

\section{Conditions on families of stable maps} \label{sec-props}
This section defines and proves basic results about the many
conditions discussed in Subsection~\ref{subsec-sketch}.  Because of
the process of \emph{modification} given in the next section, it is
necessary to work with families of stable maps that are parametrized
by reducible curves of genus $0$.  This is the setting in which all
definitions are made.

\

Let $\zeta = ((\Sigma,\sigma_1,\dots,\sigma_r),g:\Sigma \rightarrow
X)$ be an $r$-pointed genus $0$ stable map to $X$.

\begin{defn}  \label{defn-verystable}
The stable map $\zeta$ is \emph{very stable} if the prestable map
$(\Sigma,g:B\rightarrow X)$ is stable.
\end{defn}

Let $X$ be a quasi-projective scheme.  Let $B$ be a scheme.  Let
$\zeta:B \rightarrow \Kgnb{g,r}{X,e}$ be a $1$-morphism.

\begin{notat} \label{not-1mor}
The $1$-morphism $\zeta:B\rightarrow \Kgnb{g,r}{X,e}$ consists of a
datum,
\begin{equation*}
\zeta = \lt( \lt( p_\zeta:\Sigma_\zeta \rightarrow B,
\sigma_{(\zeta,1)},\dots, \sigma_{(\zeta,r)} \rt), g_\zeta:
\Sigma \rightarrow X \rt),
\end{equation*}
where
\begin{enumerate}

\item[(i)]
$p_\zeta:\Sigma_\zeta\rightarrow B$ is a proper, flat family of
connected prestable curves of arithmetic genus $g$,

\item[(ii)]
$\sigma_{(\zeta,i)}:B\rightarrow \Sigma_\zeta$, $i=1,\dots,r$ is a
collection of everywhere disjoint sections with image in the smooth
locus of $p_\zeta$, and

\item[(iii)] 
$g_\zeta:\Sigma_\zeta\rightarrow X$ is a morphism of schemes,

\end{enumerate}
that satisfies the stability condition in
Definition~\ref{defn-prestab}.  Denote by $h_{\zeta,i}:B \rightarrow
X$ the composition $g_\zeta\circ \sigma_{(\zeta,i)}$.  When there is
no risk of confusion, the subscript $\zeta$ will be omitted.
\end{notat}

Let $X$ be a smooth, quasi-projective variety.  Let $\pi:B \rightarrow
T$ be a proper, flat family of connected prestable curves of
arithmetic genus $0$.  Let $\zeta:B \rightarrow \Kgnb{0,1}{X, 1}$ be a
$1$-morphism $((p:\Sigma \rightarrow B,\sigma),f:\Sigma \rightarrow
X)$.

\begin{defn}~\label{defn-twisting}
The $1$-morphism $\zeta:B\rightarrow \Kgnb{0,1}{X,1}$ is
\emph{twisting} (resp. \emph{very twisting}) if,
\begin{enumerate}

\item[(i)] 
$(\pi:B\rightarrow T, h:B \rightarrow X)$ is a family of stable maps
to $X$, i.e., a $1$-morphism $\xi:T\rightarrow \Kgnb{0,0}{X,e}$ for
some $e\geq 0$,

\item[(ii)] 
the dimension of the obstruction group of $\Kgnb{0,0}{X,e}$ at each
point of $\xi(T)$ is $0$,

\item[(iii)] 
the dimension of the relative obstruction group of the evaluation
morphism $\text{ev}:\Kgnb{0,1}{X,1} \rightarrow X$ at each point of
$\zeta(T)$ is $0$,

\item[(iv)] 
denoting by $T_{\text{ev}}$ the dual of the sheaf of relative
differentials $\Omega_{\text{ev}}$, the pullback
$\zeta^*T_{\text{ev}}$ is $\pi$-relatively generated by global
sections (resp. $\pi$-relatively deformation ample), and

\item[(v)] 
denoting by $\text{pr}:\Kgnb{0,1}{X,1}\rightarrow \Kgnb{0,0}{X,1}$ the
projection map, and denoting by $T_{\text{pr}}$ the dual of the sheaf
of relative differentials $\Omega_{\text{pr}}$, the pullback
$\zeta^*T_{\text{pr}}$ is $\pi$-relatively generated by global
sections, i.e., the line bundle $\sigma^*\OO_\Sigma(\sigma(B))$ is
$\pi$-relatively generated by global sections.

\end{enumerate}
\end{defn}

\begin{rmk}~\label{rmk-twisting} 
\begin{enumerate} 

\item[(i)] 
The sheaf $\zeta^* T_{\text{pr}}$ is canonically isomorphic to
$\sigma^*\OO_{\Sigma}(\sigma(B))$.
  
\item[(ii)] 
The product morphism $(p,g):\Sigma\rightarrow B \times X$ is a closed
immersion whose ideal sheaf is everywhere locally defined by a regular
sequence, i.e., it is a regular embedding.  Denote by ${\mc N}$ the
normal bundle of this regular embedding.  Then (iii) of
Definition~\ref{defn-twisting} is equivalent to the condition that
$R^1p_*\lt({\mc N}(-\sigma(B))\rt) = \{0\}$.  Under this hypothesis,
$\zeta^*T_{\text{ev}}$ is canonically isomorphic to $p_*\lt({\mc
N}(-\sigma(B))\rt)$.

\item[(iii)] 
Condition (ii) is superfluous.  Since the prestable family of maps
$(\pi:B \rightarrow T, \xi:B \rightarrow X)$ is stable, also $(\pi:B
\rightarrow T, \zeta: B \rightarrow \Kgnb{0,1}{X,1})$ is stable by
Lemma~\ref{lem-XtoY} (i).
  
\item[(iv)] 
The conditions in Definition~\ref{defn-twisting} impose some
restrictions on the degrees of the locally free sheaves involved.  By
~\cite[Lem. 2.2.2]{QDiv}, the total degree of
$\sigma^*\OO_{\Sigma}(\sigma(B))$ is simply $\delta=2e-e'$ where $e$
is the degree of $h:B \rightarrow \PP^N$ and $e'$ is the degree of
$f:\Sigma \rightarrow \PP^N$.  If $\zeta$ is twisting then $2e \geq
e'$.
  
\end{enumerate}
\end{rmk}

Let $B$ be a prestable, connected curve of arithmetic genus $0$.  Let
$B_1, B_2 \subset B$ be connected subcurves of $B$ such that $B_1 \cap
B_2 = \{q\}$ is a single node, and $B=B_1 \cup B_2$.  Let $\zeta:B
\rightarrow \Kgnb{0,1}{X,1}$ be a $1$-morphism.

\begin{lem} \label{lem-twisttogether}
If $\zeta|_{B_i}:B_i \rightarrow \Kgnb{0,1}{X,1}$ is twisting for
  $i=1,2$, then $\zeta$ is twisting.  If both $\zeta_1, \zeta_2$ are
  twisting, and if at least one of them is very twisting, then $\zeta$
  is very twisting.
\end{lem}

\begin{proof}
  This follows from Lemma~\ref{lem-DAcrit}.
\end{proof}

Let $\pi:B \rightarrow T$ be a proper, flat family of connected,
prestable curves of arithmetic genus $0$.  Let $\zeta:B \rightarrow
\Kgnb{0,1}{X,1}$ be a $1$-morphism.  For every morphism of schemes
$g:T' \rightarrow T$, denote by $\pi_{T'}:B_{T'} \rightarrow T'$ the
base-change of $\pi$ by $g$, and denote by $\zeta_{T'}:B_{T'}
\rightarrow \Kgnb{0,1}{X,1}$ the composition of $B_{T'} \rightarrow B$
with $\zeta$.

\begin{lem} \label{lem-twistopen}
There is an open subscheme $U_{\text{twist}}\subset T$
(resp. $U_{vtwist}\subset T$) with the following property: for every
morphism of schemes $g:T'\rightarrow T$, the pullback family
$\pi_{T'}:B_{T'} \rightarrow T'$ and $\zeta_{T'}: B_{T'} \rightarrow
\Kgnb{0,1}{X,1}$ is twisting (resp. very twisting) iff $g(T')\subset
U_{\text{twist}}$ (resp.  $f(T')\subset U_{\text{vtwist}}$).
\end{lem}

\begin{proof}
By \cite[Lemma 1]{B} there is an open subscheme $U_1\subset T$ with
the property that for every morphism $g$, $(\pi_{T'}:B_{T'}
\rightarrow T', h_{T'}:B_{T'} \rightarrow X)$ is a family of stable
maps iff $g(T') \subset U_1$.  For every morphism $g$ such that
$\zeta_{T'}$ is twisting, $g(T') \subset U_1$.  Hence the lemma for
$\zeta_{U_1}$ implies the lemma for $\zeta$.  After replacing $T$ by
$U_1$, the morphism $h:B \rightarrow X$ is a family of stable maps
over $T$, i.e., a $1$-morphism $\xi:T \rightarrow \Kgnb{0,0}{X,e}$ for
some integer $e$.

\

Because $\pi$ has relative dimension $1$, the cohomology sheaf
$\mc{H}^k(\mathbb{R}\pi_*(L_h^\vee)) = \{0\}$ for $k\geq 2$.  By
cohomology and base change, for every geometric point $t$ of the
support of the sheaf $\mc{H}^1(\mathbb{R}\pi_*(L_h^\vee))$,
$\text{dim}\mathbb{H}^1(B_t,L_{h_t}^\vee) > 0$, i.e. the obstruction
group of $\xi_t$ does not vanish.  Therefore the dimension of the
obstruction group of $\Kgnb{0,0}{X,e}$ at each point of $\xi_{T'}(T')$
is $0$ iff $g(T')$ is contained in the complement of the support of
$\mc{H}^1(\mathbb{R}\pi_*(L_h^\vee))$.  The complement of the support
of this sheaf is an open subset of $T$.  After replacing $T$ by this
open set, the dimension of the obstruction group of $\Kgnb{0,0}{X,e}$
at each point of $\xi(T)$ is $0$.

\

For similar reasons, $T$ can be replaced by the complement of
\begin{equation*}
\pi[\text{Supp}\lt( R^1p_*({\mc N}(-\sigma(B))) \rt) ], 
\end{equation*}
where $\mc{N}$ is as in (ii) of Remark~\ref{rmk-twisting}.  After
replacing $T$ by this open subset, for every $g$, $\zeta_{T'}$
satisfies (i), (ii), and (iii) of Definition~\ref{defn-twisting}.  In
order that (v) is satisfied, it is necessary and sufficient that
$g(T')$ is contained in the complement of the support of the sheaf
\begin{equation*}
\text{Coker}
\lt( \pi^*\pi_* \sigma^*\OO_{\Sigma}(\sigma) \rightarrow
\sigma^*\OO_{\Sigma}(\sigma) \rt).
\end{equation*}
After replacing $T$ by the complement of the support of this sheaf,
for every $g$, $\zeta_{T'}$ satisfies (i), (ii), (iii) and (v) of
Definition~\ref{defn-twisting}.

\

Define $U_{\text{twist}}$ to be the complement in $T$ of the image
under $\pi$ of the support of
\begin{equation*}
\text{Coker}
\lt( \pi^*\pi_* \zeta^*T_{\text{ev}}\rightarrow \zeta^*T_{\text{ev}} \rt).
\end{equation*}
For every morphism $g$, $(\zeta^*T_{\text{ev}})_{T'}$ is
$\pi_{T'}$-relatively generated by global sections iff $g(T') \subset
U_{\text{twist}}$.  Therefore, for every morphism $g$, $\zeta_{T'}$ is
twisting iff $g(T') \subset U_{\text{twist}}$.  By
Lemma~\ref{lem-opDA}, there exists an open subset $U_{\text{vtwist}}
\subset T$ such that for every $g$, $(\zeta^* T_{\text{ev}})_{T'}$ is
$\pi_{T'}$-relatively deformation ample iff $g(T') \subset
U_{\text{vtwist}}$.  Therefore $\zeta_{T'}$ is very twisting iff
$g(T') \subset U_{\text{vtwist}}$.
\end{proof}

Let $(\pi: B \rightarrow T, h:B \rightarrow X)$ be a family of genus
$0$ stable maps, i.e., a $1$-morphism $\xi:T \rightarrow
\Kgnb{0,0}{X,e}$ for some $e \geq 0$.  For every morphism of schemes
$g:T' \rightarrow T$, denote by $\pi_{T'}:B_{T'} \rightarrow T'$ the
base-change of $\pi$ by $g$, and denote by $\xi_{T'}:B_{T'}
\rightarrow \Kgnb{0,1}{X,1}$ the composition of $B_{T'} \rightarrow B$
with $\xi$.

\begin{defn} \label{defn-twistable}
The family of stable maps $\xi:T \rightarrow \Kgnb{0,0}{X,e}$ is
\emph{twistable} (resp. \emph{very twistable}) if there exists a
surjective \'etale morphism $u:T'\rightarrow T$ and a morphism
$\zeta:B_{T'} \rightarrow \Kgnb{0,1}{X,1}$ such that $\xi_\zeta =
\xi_{T'}$ and such that $\zeta$ is twisting (resp. very twisting).
\end{defn}

\begin{prop} \label{prop-twistopen}
There is an open subscheme $U_{\text{t-able}}\subset T$ (resp
$U_{\text{vt-able}}\subset T$) such that for every morphism of schemes
$g:T'\rightarrow T$, $\xi_{T'}$ is twistable (resp. very twistable)
iff $g(T') \subset U_{\text{t-able}}$ (resp. $g(T') \subset
U_{\text{vt-able}}$).
\end{prop}

\begin{proof}
It suffices to check that if $t\in T$ is a geometric point such that
$h_{t}:B_{t} \rightarrow X$ is twistable (resp. very twistable), then
there is an \'etale neighborhood of $t\in T$ over which $\xi$ is
twistable (resp. very twistable).  Denote by $\zeta_t:B_{t}
\rightarrow \Kgnb{0,1}{X,1}$ the twisting morphism.  Consider
$\Kgnb{0,1}{X,1}$ as a quasi-projective scheme via the Pl\"ucker and
Segre embeddings of $\GG(1,n) \times \PP^n \hookrightarrow
\PP^{\frac{n(n+1)^2}{2}-1}$.  Let $\beta$ denote the degree of the
stable map $\zeta_t$.

\

Define ${\mc M} = T \times \Kgnb{0,0}{\Kgnb{0,1}{X,1}, \beta}$, i.e.,
${\mc M}$ parametrizes pairs $(s,\zeta)$ consisting of a point $s\in
T$ together with a genus $0$ stable map $\zeta:B \rightarrow
\Kgnb{0,1}{X,1}$ of degree $\beta$.  Denote the universal stable map
over $\Kgnb{0,0}{\Kgnb{0,1}{X,1},\beta}$ by
\begin{equation*}
\begin{array}{c}
\rho : {\mc B} \rightarrow \Kgnb{0,0}{\Kgnb{0,1}{X,1},\beta}, \\
\zeta : {\mc B} \rightarrow \Kgnb{0,1}{X,1}.
\end{array}
\end{equation*}
As in Notation~\ref{not-1mor}, let $p:\Sigma \rightarrow {\mc B}$ be
the pullback by $\zeta$ of the universal curve over $\Kgnb{0,1}{X,1}$,
let $\sigma:{\mc B} \rightarrow \Sigma$ be the pullback of the
universal section, let $g:\Sigma \rightarrow X$ be the pullback of the
universal map, and let $h = g \circ \sigma$.  This gives a family of
prestable maps,
\begin{equation}
\widetilde{\xi} = \lt( \rho:{\mc B} \rightarrow \Kgnb{0,0}{\Kgnb{0,1}{X,1},
  \beta}, h: {\mc B} \rightarrow X \rt).
\end{equation}
By \cite[Lemma 1]{B} there is a maximal open substack ${\mc U}_e \subset
\Kgnb{0,0}{\Kgnb{0,1}{X,1}, \beta}$ over which $\widetilde{\xi}$ is
stable of degree $e$.  By hypothesis, $(t,\zeta_t)$ is in $T \times
{\mc U}_e$.  

\

Because $\zeta_t$ is twisting, the hypotheses of Lemma~\ref{lem-XtoY}
(iii) are satisfied where $f:X \to Y$ corresponds to
$\text{ev}:\Kgnb{0,1}{X,1} \rightarrow X$ and where $(\Sigma,g:\Sigma
\rightarrow X)$ corresponds to $(B_t,\zeta_t:B_t \rightarrow
\Kgnb{0,1}{X,1})$.  Therefore at the point $(B_t,\zeta_t:B_t
\rightarrow \Kgnb{0,1}{X,1})$, the stack ${\mc U}_e$ is smooth and the
morphism $\widetilde{\xi}: {\mc U}_e \rightarrow \Kgnb{0,0}{X,e}$ is
smooth by the Jacobian criterion.

\

Consider the 1-morphism $(1_T,\xi): T\rightarrow T \times
\Kgnb{0,0}{X,e}$.  Denote by ${\mc M}$ the 2-fibered product of the
1-morphism $(1_T,\xi)$ and the 1-morphism $(1_T,\widetilde{\xi}):
T\times {\mc U}_e \rightarrow T \times \Kgnb{0,0}{X,e}$.  The
\emph{2-fibered product} ${\mc M}$ is the stack whose objects are
triples $(t,\zeta,\theta)$ consisting of a point $t\in T$, an object
$\zeta:B \rightarrow \Kgnb{0,1}{X,1}$ of ${\mc U}_e$, and an
equivalence $\theta:\xi_t \rightarrow \widetilde{\zeta}$ of objects in
the groupoid $\Kgnb{0,0}{X,e}(\SP \kappa(t))$.  Because
$(1_T,\widetilde{\xi})$ is smooth at $(t,(B_t,\zeta_t))$, the
projection $\text{pr}_1:{\mc M} \rightarrow T$ is smooth at
$(t,(B_t,\zeta_t))$.  Hence there exists an \'etale $1$-morphism $f:M
\rightarrow {\mc M}$ from a scheme $M$ to ${\mc M}$ such that
$(t,(B_t,\zeta_t))$ is in the image of $f$ and such that the composite
morphism $\text{pr}_1\circ f: M\rightarrow T$ is smooth.  By
~\cite[Prop. 2.2.14]{Ner}, there exists an \'etale morphism $u:(T',t')
\rightarrow (T,t)$ and a section $z:T' \rightarrow M$ such that
$f(z(t'))$ is $(t,(B_t,\zeta_t))$.  Denote by
$\zeta_{\text{pre}}:T'\rightarrow {\mc U}_e$ the composition
$\text{pr}_2\circ f \circ z$.

\

Denote by $\zeta:B' \rightarrow \Kgnb{0,1}{X,1}$ the pullback by
$\zeta_{\text{pre}}:T'\rightarrow {\mc U}_e$ of the universal stable
map.  By construction, $\widetilde{\xi}(\zeta): B' \rightarrow
\Kgnb{0,0}{X,e}$ is equivalent to $u^* \xi: u^*B \rightarrow
\Kgnb{0,0}{X,e}$.  Hence, after replacing $T'$ by a surjective,
\'etale cover, $B'=u^*B$ as $T'$-schemes, and $\widetilde{\xi}(\zeta)
= u^*\xi$.  The fiber of $\zeta:u^*B\rightarrow \Kgnb{0,1}{X,1}$ over
every preimage of $(t,(B_t,\zeta_t))$ is twisting.  So by
Lemma~\ref{lem-twistopen}, after replacing $T'$ by a Zariski open
subscheme whose image contains $t$, $\zeta:u^*B \rightarrow
\Kgnb{0,1}{X,1}$ is twisting.  Similarly, if $(t,(B_t,\zeta_t))$ is
very twisting, after replacing $T'$ by a Zariski open subscheme whose
image contains $t$, $\zeta$ is very twisting.  On the Zariski open
subscheme of $T$ that is the image of $u:T'\rightarrow T$, the family
$\xi:B \rightarrow \Kgnb{0,0}{X,e}$ is twistable (resp. very
twistable).
\end{proof}

Let
\begin{equation*}
\xi_i = \lt( \lt( \pi_i:B_i \rightarrow T, \sigma_i:T \rightarrow B_i \rt),
h_i:B_i \rightarrow X \rt), i=1,2,
\end{equation*}
be two families of $1$-pointed stable maps 
such that $h_1\circ \sigma_1 = h_2 \circ \sigma_2$.
Denote by
\begin{equation*}
\xi =  \lt( \pi:B \rightarrow T, h:B \rightarrow X \rt)
\end{equation*}
the family obtained by taking $B$ to be the union of $B_1$ and $B_2$
where the section $\sigma_1$ is identified with the section
$\sigma_2$.

\begin{lem} \label{lem-twisttogether2}
If for $i=1,2$ the family of unpointed stable maps
$(\pi_i:B_i\rightarrow T, h_i:B_i \rightarrow X)$ is twistable, and if
for every geometric point $t\in T$, the variety parametrizing free
lines that contain $h(\sigma(t))$ is irreducible, then $\xi$ is
twistable.  If also one of $\xi_1, \xi_2$ is very twistable, then
$\xi$ is very twistable.
\end{lem}

\begin{proof}
This follows from Lemma~\ref{lem-twisttogether}.  First of all, using
Proposition~\ref{prop-twistopen}, it suffices to prove the result when
$T=\SP k$ for some algebraically closed field $k$.
  
\

For each of $i=1,2$, let ${\mc M}_i$ denote the 2-fibered product
constructed in the proof of Proposition~\ref{prop-twistopen}, i.e.,
the objectst of ${\mc M}_i$ are pairs $(\zeta_i,\theta_i)$ consisting
of a twisting family (resp. very twisting family), $\zeta_i:B_i
\rightarrow \Kgnb{0,1}{X,1}$, such that the induced map
\begin{equation*}
\widetilde{\zeta}_i = \lt( (B_i,\sigma_i),g_i\circ \rho_i:B_i
\rightarrow X \rt)
\end{equation*}
is stable, and an equivalence of objects, $\theta_i:\xi_i \rightarrow
\widetilde{\zeta}_i$.  Since each of $\xi_i$ is twistable, each of
${\mc M}_i$ is nonempty.

\

By the proof of Proposition~\ref{prop-twistopen}, each of ${\mc M}_i$
is smooth.  By the definition of twisting families, for each $i=1,2$
the morphism
\begin{equation*}
e_i: {\mc M}_i \rightarrow \Kgnb{0,1}{X,1},\ \zeta_i\mapsto
\zeta_i(\sigma_i) 
\end{equation*}
has image contained in the unobstructed locus of $\text{ev}:
\Kgnb{0,1}{X,1}\rightarrow X$.  Let $P\subset \Kgnb{0,1}{X,1}$ denote
the preimage under $\text{ev}$ of the point
$p=h_1(\sigma_1)=h_2(\sigma_2)$.  The image of $e_i$ is contained in
the smooth locus of $P$.  The claim is that $e_i:{\mc M}_i \rightarrow
P$ is smooth.  The obstruction space at a point $\zeta_i$ is a
quotient of the cohomology group $H^1\lt( B_i, \zeta_i^*
T_{\text{ev}}(-\sigma_i) \rt)$.  By Definition ~\ref{defn-twisting}
(iv), $\zeta_i^* T_{\text{ev}}$ is generated by global sections.
Hence, by Lemma~\ref{lem-gend} (ii), this cohomology group is zero,
and $e_i$ is smooth.

\

Since both $e_1:{\mc M}_1 \rightarrow P$ and $e_2:{\mc M}_2
\rightarrow P$ are smooth, both have nonempty, open image contained in
the locus of free lines.  By hypothesis, the open subset of $P$
parametrizing free lines is irreducible.  Therefore the image of $e_1$
and the image of $e_2$ intersect.  Choose a family $\zeta_1\in {\mc
M}_1$ and $\zeta_2\in {\mc M}_2$ such that
$e_1(\zeta_1)=e_2(\zeta_2)$.  Then $\zeta_1$ and $\zeta_2$ can be
glued to obtain a morphism $\zeta:B \rightarrow \Kgnb{0,1}{X,1}$ such
that $\zeta|_{B_1}=\zeta_1$ and $\zeta|_{B_2} = \zeta_2$.  By
Lemma~\ref{lem-twisttogether}, $\zeta$ is twisting.  Moreover, if one
of $\zeta_1, \zeta_2$ is very twisting, then $\zeta$ is very twisting.
And $\widetilde{\zeta} = \xi$.  Therefore $\xi$ is twistable, and it
is very twistable if one of $\xi_1, \xi_2$ is very twistable.
\end{proof}

\begin{hyp} \label{hyp-2}
Let $U \subset \Kgnb{0,1}{X,1}$ denote the preimage of
$U_{\text{t-able}} \subset \Kgnb{0,0}{X,1}$ under $\text{pr}$.  The
evaluation morphism $\text{ev}:U \rightarrow X$ has Zariski dense
image, i.e., a general point of $X$ is contained in a twistable line.
\end{hyp}

Let $\pi:B \rightarrow T$ be a family of prestable curves of
arithmetic genus $0$.

\begin{defn} \label{defn-pos}
A $1$-morphism $\zeta:B \rightarrow \Kgnb{0,1}{X,e}$ is
\emph{positive} (resp. \emph{very positive}) if
\begin{enumerate}

\item[(i)] 
$(\pi:B \rightarrow T, h:B \rightarrow X)$ is a family of stable maps,
i.e., a 1-morphism $\xi:T \rightarrow \Kgnb{0,0}{X,\epsilon}$ for some
$\epsilon\geq 0$,

\item [(ii)]
the dimension of the obstruction group of $\Kgnb{0,0}{X,\epsilon}$ is
$0$ at each point of $\xi(T)$,

\item[(iii)] 
the dimension of the obstruction group of $\Kgnb{0,0}{X,e}$ is $0$ at
each point of $\text{pr}(\zeta(T))$,

\item[(iv)] 
the pullback $(\text{pr}\circ \zeta)^* T_{\Kgnb{0,0}{X,e}}$ is
$\pi$-relatively deformation ample, and

\item[(v)] 
the pullback $\sigma^*\OO_{\Sigma}(\sigma)$ is $\pi$-relatively
generated by global sections (resp. $\pi$-relatively ample).

\end{enumerate}
\end{defn}

\begin{rmk} \label{rmk-pos}
This definition is similar to Definition~\ref{defn-twisting}.  It
differs in that $e$ need not equal $1$ and that $\text{pr}\circ \zeta$
has image in the unobstructed locus, instead of requiring $\zeta$ to
have image in the unobstructed locus of the morphism $\text{ev}$.
\end{rmk}

\begin{lem}\label{lem-posopen}
Let $\pi:B \rightarrow T$ be a family of prestable curves of
arithmetic genus $0$ and let $\zeta:B \rightarrow \Kgnb{0,1}{X,1}$ be
a $1$-morphism.  There is an open subscheme $U_{\text{pos}} \subset T$
(resp. $U_{\text{v-pos}}\subset T$) with the following property: for
every morphism of schemes $f:T'\rightarrow T$, the pullback family
$f^*\pi: f^*B \rightarrow T'$ and $f^*\zeta: f^*B \rightarrow
\Kgnb{0,1}{X,1}$ is positive (resp. very positive) iff $f(T')\subset
U_{\text{pos}}$ (resp.  $f(T')\subset U_{\text{v-pos}}$).
\end{lem}

\begin{proof}
The proof is almost identical to the proof of
lemma~\ref{lem-twistopen}.
\end{proof}

\begin{lem} \label{lem-posdef}
Let $B$ be a prestable curve of arithmetic genus $0$, and let
$\zeta:B\rightarrow \Kgnb{0,1}{X,e}$ be a positive $1$-morphism whose
image is contained in the locus of very stable maps.
\begin{enumerate}

\item[(i)] 
If $B$ is smooth, then $\zeta:B \rightarrow \Kgnb{0,1}{X,e}$ is free,
i.e., $\zeta^* T_{\Kgnb{0,1}{X,e}}$ is generated by global sections.
If $\zeta$ is very positive, then $\zeta$ is very free, i.e., $\zeta^*
T_{\Kgnb{0,1}{X,e}}$ is ample.

\item[(ii)]
The dimension of the obstruction group of the 1-morphism $\zeta:B
\rightarrow \Kgnb{0,1}{X,e}$ is $0$.  In particular there is a
discrete valuation ring, $R$, a family of prestable curves,
$\pi:\mc{B} \rightarrow \SP R$, and a positive 1-morphism
$\zeta_R:\mc{B} \rightarrow \Kgnb{0,1}{X,e}$ such that the geometric
closed fiber is $\zeta$, such that the geometric generic fiber
$\mc{B}_\eta$ is smooth, and such that $\zeta_R(\mc{B})$ is contained
in the locus of very stable maps.

\end{enumerate}
\end{lem}

\begin{proof}
\textbf{(i)}: By hypothesis, $\zeta$ is positive (resp. very
positive).  By the relative version of Lemma~\ref{lem-def3a}, the
image of $\zeta$ is in the smooth locus of $\Kgnb{0,1}{X,e}$ and there
is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> \sigma^* \OO_{\Sigma}(\sigma) @>>> \zeta^* T_{\Kgnb{0,1}{X,e}}
@>>> (\text{pr}\circ \zeta)^* T_{\Kgnb{0,0}{X,e}} @>>> 0.
\end{CD}
\end{equation*}
By Definition ~\ref{defn-pos} (iv) and (v), $(\text{pr}\circ \zeta)^*
T_{\Kgnb{0,0}{X,e}}$ is deformation ample and
$\sigma^*\OO_{\Sigma}(\sigma)$ is generated by global sections
(resp. deformation ample).  Hence $\zeta^* T_{\Kgnb{0,1}{X,e}}$ is
generated by global sections (resp. deformation ample by
Lemma~\ref{lem-secDA} (ii)).  Since $B$ is smooth $\zeta$ is free
(resp. very free).


\

\textbf{(ii)}: Let $R'$ be a discrete valuation ring and let $\pi:
{\mc B}' \rightarrow \SP R'$ be a smoothing of $B$, i.e., there is an
isomorphism $i:B \rightarrow \mc{B}'_0$ and the generic fiber
$\mc{B}'_\eta$ is smooth.  For technical reasons it is necessary to
compactify $\Kgnb{0,1}{X,e}$.  Let $j:X \hookrightarrow \overline{X}$
be an open immersion of $X$ into a projective scheme, and let
$j_*:\Kgnb{0,1}{X,e} \hookrightarrow \Kgnb{0,1}{\overline{X},e}$ be
the corresponding open immersion of moduli stacks.  By ~\cite{FP},
$\Kgnb{0,1}{\overline{X},e}$ is a proper Deligne-Mumford stack with
projective coarse moduli space.  Form the $R'$-stack, $\mc{B}' \times
\Kgnb{0,1}{\overline{X},e}$.  With respect to an ample invertible
sheaf on the coarse moduli space, let $\epsilon$ denote the degree of
the map $(i,\zeta):B \rightarrow \mc{B}' \times
\Kgnb{0,1}{\overline{X},e}$.  Then $(i,\zeta)$ is an object of the
\emph{Abramovich-Vistoli stack},
\begin{equation*}
\mc{K} = \mc{K}_{0,0}(\mc{B}'\times
\Kgnb{0,1}{\overline{X},e},\epsilon) \rightarrow \SP R',
\end{equation*}
that parametrizes \emph{twisted stable maps} to $\mc{B}' \times
\Kgnb{0,1}{\overline{X},e}$ over $\SP R'$, cf. ~\cite{AV}.  The stack
$\mc{K} \rightarrow \SP R'$ is a Deligne-Mumford stack that is proper
over $\SP R'$, and the coarse moduli space is projective over $\SP
R'$.

\

The deformation theory of twisted stable maps is developed in
~\cite[Section 5]{AV}, and is exactly analogous to the deformation
theory of stable maps.  In particular, the obstruction space to the
morphism $o:\mc{K} \rightarrow \SP R'$ at $[(i,\zeta)]$ is a quotient
of $H^1(B, \zeta^* T_{\Kgnb{0,1}{X,e}})$.  Since $\zeta^*
T_{\Kgnb{0,1}{X,e}}$ is generated by global sections, by Lemma
~\ref{lem-gend} (i), $h^1(B,\zeta^* T_{\Kgnb{0,1}{X,e}}) =0$.
Therefore $o$ is smooth at $[(i,\zeta)]$.  By
~\cite[Prop. 2.2.14]{Ner}, there exists an \'etale morphism of
discrete valuation ring's, $\SP R \rightarrow \SP R'$ and a section
$s:\SP R \rightarrow \mc{K}$ of $o$.  In other words, there exists a
family of twisted stable maps over $\SP R$ whose closed fiber is
isomorphic to the pullback of $(i,\zeta)$, i.e.,
\begin{equation*}
(\pi: \mc{B} \rightarrow \SP R', \lambda: \mc{B}
\rightarrow \SP R \times_{\SP R'} (\mc{B}' \times
\Kgnb{0,1}{\overline{X},e})).
\end{equation*}   

\

Because $B$ is a scheme, the family of curves over $\SP R$ is a
scheme, i.e., $\pi:\mc{B} \rightarrow \SP R$ is a morphism of schemes.
Because the composition of $(i,\zeta)$ with projection onto the closed
fiber of $\mc{B}'$ is an isomorphism, the composition
\begin{equation*}
\mc{B} \xrightarrow{\lambda} \SP R \times_{\SP R'} (\mc{B}' \times
\Kgnb{0,1}{\overline{X},e}) \xrightarrow{\text{pr}_{\mc{B}'}} 
\SP R \times_{\SP R'} \mc{B'},
\end{equation*}
is an isomorphism.  Since the image of $(i,\zeta)$ is contained in the
open substack $\mc{B}' \times \Kgnb{0,1}{X,e}$, the image of $\lambda$
is contained in the the open substack $\SP R \times_{\SP R'} (\mc{B}'
\times \Kgnb{0,1}{X,e}$.  Putting the pieces together, $\pi:\mc{B}
\rightarrow \SP R$ is a smoothing of $B$, and $\lambda$ induces a
1-morphism of $R$-stacks, $\zeta_R: \mc{B} \rightarrow
\Kgnb{0,1}{X,e}$ such that the closed fiber is $\zeta$.

\

By Lemma ~\ref{lem-posopen}, $\zeta_R$ is positive.  Since the locus
of very stable maps in $\Kgnb{0,1}{X,e}$ is open, the image of
$\zeta_R$ is contained in this locus.
\end{proof}

\begin{rmk} \label{rmk-olsson}
The use of the Abramovich-Vistoli stack above is a bit contrived.  The
most natural stack to use is the Hom stack parametrizing 1-morphisms
between two flat, proper, tame Deligne-Mumford stacks.  Martin Olsson
has proved existence and some foundational properties of the Hom stack
in the generality needed above, but the details have not yet been
published.
\end{rmk}

Following is the main definition of this section.

\begin{defn} \label{defn-inducts}
Let $\pi:B \rightarrow T$ be a family of prestable curves of
arithmetic genus $0$.  An \emph{inducting pair of degree $e$} is a
pair of 1-morphisms
\begin{equation*}
\lt( \zeta_1:B\rightarrow \Kgnb{0,1}{X,1}, \overline{\zeta}_e:B
\rightarrow \Kgnb{0,1}{X,e} \rt),
\end{equation*}
such that
\begin{enumerate}

\item[(i)] 
$\zeta_1$ is very twisting,

\item[(ii)] 
$\overline{\zeta}_e$ is very positive and the image of
$\overline{\zeta}_e$ is contained in the locus of very stable maps,
and

\item[(iii)] 
the morphisms $h_{\zeta_1}:B \rightarrow X$ and
$h_{\overline{\zeta}_e}: B \rightarrow X$ are equal.
\end{enumerate}
\end{defn}

\begin{lem} \label{lem-inducting}
Let $\pi:B \rightarrow T$ be a family of prestable curves of
arithmetic genus $0$, and let
\begin{equation*}
\lt( \zeta_1:B
\rightarrow \Kgnb{0,1}{X,1}, \overline{\zeta}_e:B \rightarrow
\Kgnb{0,1}{X,e} \rt),
\end{equation*}
be a pair of $1$-morphisms such that $h_{\zeta_1} =
h_{\overline{\zeta}_e}$.  Then there is an open subscheme
$U_{\text{induct}}\subset T$ with the following property: for every
morphism of schemes $f:T'\rightarrow T$, the pullback of
$(\zeta_1,\overline{\zeta}_e)$ is inducting iff $f(T')\subset U$.
\end{lem}

\begin{proof}
Define $U_{\text{induct}}$ to be the intersection of the open subset
$U_{\text{vtwist}}\subset T$ from Lemma~\ref{lem-twistopen} for
$\zeta_1$ and the open subset $U_{\text{v-pos}}\subset T$ from
Lemma~\ref{lem-posopen} for $\overline{\zeta}_e$.
\end{proof}

The final definition of this section is the following.

\begin{defn} \label{defn-inductable}
Let $(\pi:B\rightarrow T, \overline{\zeta}_e:B \rightarrow
\Kgnb{0,1}{X,e})$ be a very positive family whose image is contained
in the locus of very stable maps.  The 1-morphism $\overline{\zeta}_e$
is \emph{inductable} if there exists a surjective \'etale morphism
$u:T' \rightarrow T$ and a morphism $\zeta_1:u^*B \rightarrow
\Kgnb{0,1}{X,1}$ such that
\begin{enumerate}

\item[(i)]
$h_{\zeta_1} = u^* h_{\overline{\zeta}_e}$, and

\item[(ii)]
 $(\zeta_1,\overline{\zeta}_e)$ is an inducting pair.

\end{enumerate}
\end{defn}

\begin{lem} \label{lem-inductable}
Let $(\pi:B \rightarrow T, \overline{\zeta}_e:B \rightarrow
\Kgnb{0,1}{X,e})$ be a very free family.  There is an open subscheme
$U_{i-able} \subset T$ with the following property: for every morphism
of schemes $f:T' \rightarrow T$, the pullback $(f^*\pi: f^*B
\rightarrow T', f^*\overline{\zeta}_e: f^* B \rightarrow
\Kgnb{0,1}{X,e})$ is inductable iff $f(T')\subset U_{i-able}$.
\end{lem}

\begin{proof}
  Apply Proposition~\ref{prop-twistopen} to
\begin{equation*}
\xi := (\pi:B
\rightarrow T, h_{\overline{\zeta}_e}: B \rightarrow X).
\end{equation*}
\end{proof}

\section{The induction argument} \label{sec-induct}
In this section it is proved that if $X$ satisfies Hypotheses
\ref{hyp-1}, \ref{hyp-1.5}, \ref{hyp-1.75}, and \ref{hyp-2}, and if
there exists an inductable $1$-morphism $\overline{\zeta}_e: B
\rightarrow \Kgnb{0,1}{X,e}$, then there exists an inductable
$1$-morphism $\overline{\zeta}_{e+1}: B \rightarrow
\Kgnb{0,1}{X,e+1}$.  The basic idea is, given an inducting pair
$(\zeta_1,\overline{\zeta}_{e+1})$, to form the family of ``connected
sums'', i.e., the family of reducible curves obtained by gluing the
families $\zeta_1$ and $\overline{\zeta}_{e+1}$ along the two
sections.  However, this is a family of unpointed curves, and an
inducting family is a family of pointed curves.  By hypothesis, the
section of $\overline{\zeta}_e$, $\overline{\sigma}:B \rightarrow
\overline{\Sigma}$, is such that the Cartier divisor
$\overline{\sigma}(B) \subset \overline{\Sigma}$ moves.  A general
member of the linear system $|\overline{\sigma}(B)|$ is the image of a
section, $\sigma':B \rightarrow \overline{\Sigma}$.  Except at
finitely many points of $B$, the sections $\overline{\sigma}$ and
$\sigma'$ are disjoint.  Away from these points, the section of the
family of reducible curves is taken to be $\sigma'$.  The stable limit
over the finitely many points of $B$ is obtained by blowing up
$\overline{\Sigma}$ along the zero-dimensional scheme
$\overline{\sigma}(B) \cap \sigma'(B)$.  Unfortunately, the resulting
family of pointed, reducible curves is no longer very positive.

\

To make the resulting family very positive, the family is altered by a
\emph{modification}.



\subsection{Modification} \label{subsec-mod}

\begin{defn} \label{defn-input}
An \emph{input triple} is a triple $I = (\zeta,L,\sigma_i)$ where
\begin{enumerate}

\item[(i)]
$\zeta = ((\Sigma,\sigma_1,\dots,\sigma_r),g:\Sigma \rightarrow X)$ is
an $r$-pointed stable map to $X$ of arithmetic genus $g$,

\item[(ii)]
$L \subset \Sigma$ is an irreducible component that is smooth, and

\item[(iii)]
$\sigma_i \in L$ is a marked point.  

\end{enumerate}
\end{defn}

Let $I = (\zeta,L,\sigma_i)$ be an input triple.  Denote by $M \subset
\Sigma$ the union of all irreducible components other than $L$.
Denote by $R = (\rho_1, \dots, \rho_c)$ the intersection $L \cap M$,
denote by $S = (\sigma_{j_1}, \dots, \sigma_{j_d})$ the marked points
that are contained in $L$ other than $\sigma_i$, and denote by $S' =
(\sigma_{k_1},\dots, \sigma_{k_e})$ the marked points that are
contained in $M$.

\

Denote by $\Delta: L \rightarrow L \times L$ the diagonal, and denote
by $u:\Lambda \rightarrow L \times L$ the blowing up of $L \times L$
along $\Delta(R \cup S)$.  For each closed point $\lambda \in R \cup
S$, denote by $F_\lambda \subset \Lambda$ the proper transform of $L
\times\{ \lambda\} \subset L \times L$.  Denote by $F_{\Delta} \subset
\Lambda$ the proper transform of $\Delta(L) \subset L \times L$.  For
each $\lambda \in R \cup S$, there exists a unique section of
$\text{pr}_1 \circ u: \Lambda \rightarrow L$, $\sigma_{I,\lambda}:L
\rightarrow \Lambda$, such that $F_\lambda = \sigma_{I,\lambda}(L)$.
Also there is a section of $\text{pr}_1 \circ u$, $\sigma_{I,\Delta}$,
such that $\sigma_{I,\Delta}(L) = F_{\Delta}$.

\

Consider the projection $\text{pr}_1: L \times M \rightarrow L$.  For
each closed point $\lambda \in R \cup S'$ there is a unique section of
$\text{pr}_1$, $\sigma'_{I,\lambda}$, such that
$\sigma'_{I,\lambda}(L) = L \times \{\lambda \}$.  Let $\Sigma_I$ be
the simple normal crossing surface containing $\Lambda$ and $L \times
M$ that is obtained by identifying the divisor $\sigma_{I,\lambda}(L)
\subset \Lambda$ with $\sigma'_{I,\lambda}(L) \subset L \times M$ for
each $\lambda \in R$.  The divisors are identified by
$\sigma_{I,\lambda}(t) \leftrightarrow \sigma'_{I,\lambda}(t)$ for
each $t\in L$.  There is a unique morphism $p_I: \Sigma_I \rightarrow
L$ whose restriction to $\Lambda$ is $\text{pr}_1 \circ u$ and whose
restriction to $L \times M$ is $\text{pr}_1$.  There is a unique
morphism $g_I: \Sigma_I \rightarrow X$ whose restriction to $\Lambda$
is $g|_L \circ \text{pr}_2 \circ u$ and whose restriction to $L \times
M$ is $g|_M \circ \text{pr}_2$.  For each integer $j=1,\dots,r$ such
that $j\neq i$, there exists a section of $p_I$, $\sigma_{I,j}: L
\rightarrow \Sigma_I$ given by
\begin{equation*}
\sigma_{I,j} = \lt\{ \begin{array}{ll}
                      \sigma_{I,\lambda}, & \lambda = \sigma_j \in S,
                      \\
                      \sigma'_{I,\lambda}, & \lambda = \sigma_j \in S'
                     \end{array} \rt.
\end{equation*}
And there exists a section $\sigma_{I,i}:L \rightarrow \Sigma_I$ given
by $\sigma_{I,i} = \sigma_{I,\Delta}$.  By construction, the sections
$\sigma_{I,j}$ are pairwise disjoint.  Hence the datum
\begin{equation*}
\zeta_I = ((p_I:\Sigma_I
\rightarrow L, \sigma_{I,1},\dots, \sigma_{I,r}),g_I:\Sigma_I
\rightarrow X)
\end{equation*}
is a family of $r$-pointed stable maps to $X$ of arithmetic genus $g$.
And the fiber over $\sigma_i \in L$ is canonically isomorphic to
$\zeta$.

\begin{defn} \label{defn-modif1}
The family of stable maps, $\zeta_I$, is the \emph{modification
associated to $I$}.
\end{defn}

The relevance of this construction is the following.  Let $B$ be a
prestable curve and let
\begin{equation*}
\zeta = ((p:\Sigma \rightarrow
B,\sigma_1,\dots,\sigma_r), g:\Sigma \rightarrow X)
\end{equation*}
be a family of $r$-pointed stable maps to $X$ of arithmetic genus $g$.
Let $b\in B$ be a smooth point, let $L \subset \Sigma_b$ be an
irreducible component that is smooth, and let $\sigma_i(b) \in L$ be a
marked point.  Then $I = (\zeta_b,L,\sigma_i(b))$ is an input triple.
Let $\zeta_I$ be the modification associated to $I$.  The fiber
$\zeta_{I,\sigma_i(b)}$ is canonically isomorphic to the fiber
$\zeta_b$.  Let $\widetilde{B}$ be the prestable curve containing $B$
and $L$ obtained by identifying $b\in B$ with $\sigma_i(b) \in L$.
There is a unique family of $r$-pointed stable maps to $X$ of
arithmetic genus $g$,
\begin{equation*}
\widetilde{\zeta} = ((\widetilde{p}:\widetilde{\Sigma} \rightarrow
\widetilde{B}, \widetilde{\sigma}_1,\dots, \widetilde{\sigma}_r),
\widetilde{g}: \widetilde{\Sigma} \rightarrow X),
\end{equation*}
whose restriction to $B$ is $\zeta$ and whose restriction to $L$ is
$\zeta_I$.  

\begin{defn} \label{defn-modif2}
The family of stable maps, $\widetilde{\zeta}$, is the
\emph{modification of $\zeta$ associated to $(b,L,\sigma_i(b))$}.
\end{defn}

Let $I = (\zeta,L,\sigma_i)$ be an input triple.  Denote by
$\zeta_{\widehat{\sigma_i}}$ the $(r-1)$-pointed stable map obtained
by forgetting the marked point $\sigma_i$.

\begin{lem} \label{lem-modif1}
Let $I = (\zeta,L,\sigma_i)$ be an input triple.
\begin{enumerate}

\item[(i)]
The dimension of the obstruction space of $\zeta_{\widehat{\sigma_i}}$
is $0$ and for each closed point $\lambda \in R$, there exists a
first-order deformation of $\zeta_{\widehat{\sigma_i}}$ that smoothes
the node $\lambda$ iff the same is true for $\zeta$.

\item[(ii)]
For every closed point $\lambda \in L$, the dimension of the
obstruction space of $\zeta_{I,\lambda}$ is $0$ iff both the dimension
of the obstruction space of $\zeta_{\widehat{\sigma_i}}$ is $0$ and
for each closed point $\lambda \in R$ there exists a first-order
deformation of $\zeta_{\widehat{\sigma_i}}$ that smoothes the node
$\lambda$.

\item[(iii)]
If the equivalent conditions of (ii) are satisfied then there is an
exact sequence,
\begin{equation*}
\begin{array}{ccccccc}
0 & \rightarrow & T_L & \rightarrow & \zeta_I^* T_{\Kgnb{g,r}{X,e}} &
\rightarrow & \dots \\
\dots & \rightarrow & 
T_{\Kgnb{g,r-1}{X,e},\zeta_{\widehat{\sigma_i}}} \otimes_{\CC} \OO_L 
& \rightarrow &
\bigoplus_{\lambda \in R} T_{L,\lambda} \otimes_{\OO_\lambda}
T_{M,\lambda} & \rightarrow & 0.
\end{array}
\end{equation*}

\item[(iv)]
If the equivalent conditions of (iii) are satisfied, if $L$ has genus
$0$, and if the map,
\begin{equation*}
T_{\Kgnb{g,r-1}{X,e},\zeta_{\widehat{\sigma_i}}}
\rightarrow  \oplus_{\lambda \in R} T_{L,\lambda} \otimes
T_{M,\lambda},
\end{equation*}
is surjective, then $h^1(L,\zeta_I^* T_{\Kgnb{g,r}{X,e}}) = 0$.  

\end{enumerate}
\end{lem}

\begin{proof}
The item (i) follows by Lemma~\ref{lem-def3b}.  The items (ii) and
(iii) follow by Lemmas ~\ref{lem-def2a}, ~\ref{lem-def2b},
~\ref{lem-def3a} and ~\ref{lem-def3b}.  If $L$ has genus $0$, then
$T_L$ is generated by global sections.  So (iv) follows from the long
exact sequences in cohomology associated to the exact sequence in (ii)
and by Lemma~\ref{lem-gend} (i).
\end{proof}

Let $\zeta$ be a stable map of arithmetic genus $0$ and let $I =
(\zeta,L,\sigma_i)$ be an input pair.

\begin{lem} \label{lem-modif2}
If the dimension of the obstruction group of $g|_L:L \rightarrow X$ is
$0$, and if the pullback of $T_X$ by $g|_M:M \rightarrow X$ is
generated by global sections, then for every closed point $\lambda \in
L$, the dimension of the obstruction space of $\zeta_{I,\lambda}$ is
$0$, and $h^1(L,\zeta_I^* T_{\Kgnb{0,r}{X,e}}) = 0$.
\end{lem}

\begin{proof}
By Lemma~\ref{lem-def1b}, Lemma~\ref{lem-def3a} and
Lemma~\ref{lem-def3b}, the family of pointed stable curves,
\begin{equation*}
((\text{pr}_1 \circ u: \Lambda \rightarrow L, \sigma_{\lambda},
\sigma_{\Delta}), g|_L \circ \text{pr}_2 \circ u:\Lambda \rightarrow
X),
\end{equation*}
is such that for each point of $L$ the dimension of the obstruction
space is $0$.  Consider the families obtained by successively gluing
on the connected components of $L \times M$.  Applying
Lemma~\ref{lem-glue2} to each of these families, the hypotheses of
Lemma~\ref{lem-modif1} (iv) are satisfied for $\zeta_I$.
\end{proof} 

Let $B$ be a prestable curve of arithmetic genus $0$, and let
\begin{equation*}
\zeta = ((p:\Sigma \rightarrow B,\sigma_1,\dots,\sigma_r),g:\Sigma
\rightarrow X),
\end{equation*}
be a family of $r$-pointed stable maps of arithmetic genus $0$.  Let
$b\in B$ be a smooth point, let $L \subset \Sigma_b$ be an irreducible
component that is not contracted by $g$, and let $\sigma_i(b) \in L$
be a marked point. Let $M$ be the union of all irreducible components
of $\Sigma_b$ other than $L$.

\begin{lem} \label{lem-modif3}
If for every point $b'\in B$ the dimension of the obstruction group of
$\zeta_{b'}$ is $0$, if $\zeta^* T_{\Kgnb{0,r}{X,e}}$ is generated by
global sections, if the dimension of the obstruction group of $g|_L:L
\rightarrow X$ is $0$, and if the pullback of $T_X$ by $g|_M:M
\rightarrow X$ is generated by global sections, then for every point
$b' \in \widetilde{B}$ the dimension of the obstruction group of
$\widetilde{\zeta}_{b'}$ is $0$ and $h^1(\widetilde{B},
\widetilde{\zeta}^* T_{\Kgnb{0,r}{X,e}})=0$.

Therefore the dimension of the obstruction group of
$\widetilde{\zeta}:\widetilde{B} \rightarrow \Kgnb{0,r}{X,e}$ is $0$.
In particular there is a discrete valuation ring, $R$, a family of
prestable curves, $\pi:\mc{B} \rightarrow \SP R$, and a 1-morphism
$\zeta_R:\mc{B} \rightarrow \Kgnb{0,r}{X,e}$ such that the geometric
closed fiber is $\widetilde{\zeta}$ and such that the geometric
generic fiber $\mc{B}_\eta$ is smooth.
\end{lem}

\begin{proof}
The hypotheses of Lemma~\ref{lem-modif2} are satisfied for the
modification associated to $I=(\zeta_b,L,\sigma_i(b))$,
$\widetilde{\zeta}_L:L \rightarrow \Kgnb{0,r}{X,e}$.  Hence the
dimension of the obstruction group of $\widetilde{\zeta}$ at each
closed point of $L$ is $0$ and $h^1(L,\widetilde{\zeta}_L^*
T_{\Kgnb{0,r}{X,e}})=0$.  There is a short exact sequence of coherent
sheaves on $\widetilde{B}$,
\begin{equation*}
\begin{CD}
0 @>>> \zeta^* T_{\Kgnb{0,r}{X,e}}(-b) @>>> \widetilde{\zeta}^*
T_{\Kgnb{0,r}{X,e}} @>>> \widetilde{\zeta}_L^* T_{\Kgnb{0,r}{X,e}}
@>>> 0.
\end{CD}
\end{equation*}
By Lemma~\ref{lem-gend} (ii), $h^1(B, \zeta^*T_{\Kgnb{0,r}{X,e}}(-b))
= 0$.  Hence by the long exact sequence of cohomology associated to
the short exact sequence above, $h^1(\widetilde{B},
\widetilde{\zeta}^* T_{\Kgnb{0,r}{X,e}}) = 0$.

\

The second part of the lemma follows by the same proof as in
Lemma~\ref{lem-posdef} (ii).
\end{proof}

\subsection{The induction argument} \label{subsec-induct}

\begin{notat} \label{not-divs}
The notation for the generators of the group of $\QQ$-Cartier divisor
classes on $\Kgnb{0,1}{\PP^n,e}$ is taken from ~\cite{QDiv}.

\begin{enumerate}

\item[(i)]
For each integer $0 \leq i \leq \lfloor e \rfloor$ denote by
$\Delta_i$, or sometimes $\Delta_{i,e-i}$, the class of the
$\QQ$-Cartier divisor that is the closure of the locus parametrizing
embedded curves that have one irreducible component of degree $i$, one
irreducible component of degree $e-i$ and where the marked point is on
the irreducible component of degree $i$.

\item[(ii)]
Denote by $\mc{L}$ the first Chern class of $\text{ev}^*
\OO_{\PP^n}(1)$.

\item[(iii)]
Denote by $\mc{H}$ the class of the $\QQ$-Cartier divisor
parametrizing stable maps $g:\Sigma \rightarrow \PP^n$ such that
$g(\Sigma) \cap \Lambda \neq \emptyset$, where $\Lambda \subset \PP^n$
is a fixed linear $\PP^{n-2}$.

\end{enumerate}
\end{notat}

Let $B$ be a prestable curve of genus $0$.  Let $\zeta = ((p:\Sigma
\rightarrow B, \sigma),g:\Sigma \rightarrow X)$ be a family of stable
maps of genus $0$ and degree $e>1$.  Denote by $h:B \rightarrow X$ the
composition $g \circ \sigma$.

\begin{defn} \label{defn-typical}
The $1$-morphism $\zeta:B \rightarrow \Kgnb{0,1}{X,e}$ is
\emph{typical} if,
\begin{enumerate}

\item[(i)]
the curve $B$ is smooth,

\item[(ii)]
for every closed point $b\in B$, the obstruction group of $\zeta_b$
has dimension $0$,

\item[(iii)] 
for every point $b\in B$, the stable map $\zeta_b$ has only the
trivial automorphism,

\item[(iv)] 
for every integer $j=1,\dots, e-1$, the preimage
$\zeta^{-1}(\Delta_j)$ consists of finitely many points, and for each
$b \in \zeta^{-1}(\Delta_j)$, the curve $\Sigma_b$ has only two
irreducible components,

\item[(v)]
for every point $b\in \zeta^{-1}(\Delta_1)$, $\Sigma_b$ is a union of
$2$ irreducible components $L \cup M$ with $\sigma(b) \in L$ such that
%the node of $L \cup M$ maps under $h$ to a general point of $X$, and
$g|_L:L \rightarrow X$ is an isomorphism to a twistable line.
\end{enumerate}
\end{defn}

\begin{lem} \label{lem-typical1}
Let $\pi:B \rightarrow T$ be a family of prestable curves of
arithmetic genus $0$ and let $\zeta:B \rightarrow \Kgnb{0,1}{X,e}$ be
a $1$-morphism.  There is an open subscheme $U_{\text{typ}} \subset T$
with the following property: for every morphism of schemes $f:T'
\rightarrow T$, the pullback family $f^*\pi: f^*B \rightarrow T'$ and
$f^*\zeta: f^* B \rightarrow \Kgnb{0,1}{X,e}$ is typical iff $f(T')
\subset U_{\text{typ}}$.
\end{lem}

\begin{proof}
Each of the conditions in Definition~\ref{defn-typical} is clearly an
open condition.
\end{proof}

\begin{lem} \label{lem-typical2}
If $X$ satisfies Hypotheses ~\ref{hyp-1}, ~\ref{hyp-1.5},
~\ref{hyp-1.75}, and ~\ref{hyp-2}, if for every closed point $b\in B$
the obstruction group of $\zeta_b$ has dimension $0$, and if $\zeta^*
T_{\Kgnb{0,1}{X,e}}$ is generated by global sections, then there
exists a discrete valuation ring, $R$, a family of prestable curves,
$\pi:\mc{B} \rightarrow \SP R$, and a $1$-morphism $\zeta_R: \mc{B}
\rightarrow \Kgnb{0,1}{X,e}$ such that the geometric closed fiber is
$\zeta$, such that the geometric generic fiber $\mc{B}_\eta$ is
smooth, and such that $(\zeta_R)_\eta$ is typical.
\end{lem}

\begin{proof}
By the same argument as in the proof of Lemma~\ref{lem-posopen}, there
exists a family $\zeta_R: \mc{B} \rightarrow \Kgnb{0,1}{X,e}$ such
that $\mc{B}_\eta$ is smooth.  The condition that for every point the
obstruction group has dimension $0$ is stable under generization, so
it also holds for $(\zeta_R)_\eta$.  Similarly, $(\zeta_R)_\eta^*
T_{\Kgnb{0,1}{X,e}}$ is generated by global sections.  By
~\cite[Prop. 3.7]{K}, the family $\zeta_R$ may be chosen so that
$(\zeta_R)(B_\eta)$ is disjoint from any given finite collection of
closed substacks of codimension at least $2$.  By
Hypothesis~\ref{hyp-1}, the locus of stable maps such that the domain
curve has $3$ or more irreducible components has codimension at least
$2$.  Let $Z \subset \Delta_i$ be a closed substack whose image in the
coarse moduli space of $\Delta_i$ is an ample divisor.  Again by
Hypothesis~\ref{hyp-1}, $Z$ has codimension $2$ in $\Kgnb{0,1}{X,e}$.
So $(\zeta_R)(B_\eta)$ does not intersect $Z$, and therefore it is not
contained in $\Delta_i$.  By Hypothesis~\ref{hyp-1.75}, the locus of
stable maps that have a nontrivial automorphism has codimension at
least $2$.  Finally, by Hypothesis ~\ref{hyp-1.5} and
Hypothesis~\ref{hyp-2}, the locus in $\Delta_1$ parametrizing stable
maps $g:L \cup M \rightarrow X$ such that $g|_L:L \rightarrow X$ is an
isomorphism to a twistable line is a dense open in $\Delta_1$.  In
particular the complement in $\Delta_1$ is a closed substack that has
codimension $2$ in $\Kgnb{0,1}{X,e}$.  Therefore $(\zeta_R)_\eta$ is
typical.
\end{proof}

By Hypothesis ~\ref{hyp-2}, a general line $L\subset X$ is twistable.
Denote by,
\begin{equation*}
\zeta_L = ((p_L:\Sigma_L \rightarrow L,\sigma_L),g_L:\Sigma_L
\rightarrow X),
\end{equation*}
a twisting family such that $h_L:L \rightarrow X$ is the inclusion.
By Remark ~\ref{rmk-twisting} (iv), the degree of $g_L$ is either $1$
or $2$, i.e., either $g_L:\Sigma_L \rightarrow X$ is a birational map
to a linear $\PP^2$ in $X$ (obtained by blowing up a point on the
$\PP^2$), or $g_L$ is an isomorphism of $\Sigma_L$ to a nonsingular
quadric surface in $X$.

\begin{defn} \label{defn-cases}
Let $X$ be a variety that satisfies Hypotheses ~\ref{hyp-1.5} and
~\ref{hyp-2}.  The variety $X$ is \emph{planar type} if for a general
line $L$ there exists a twisting family such that the degree of $g_L$
is $1$.  The variety is \emph{quadric type} if it is not planar type.
\end{defn}

\begin{rmk} \label{rmk-cases}
It is not hard to prove that if $X$ is planar type, then there exists
an integer $r \geq 2$ such that for a general point $x\in X$, the set
of lines in $X$ that contain $x$ sweep out a linear $\PP^r$.  Let $Q$
denote the irreducible component of the Hilbert scheme of $\PP^r$'s in
$X$ that contains the $\PP^r$'s constructed in this way.  Then $X$ is
birationally a $\PP^r$-bundle over $Q$ and, with its natural Pl\"ucker
embedding, $Q$ is not uniruled by lines.  If $X$ satisfies Hypothesis
~\ref{hyp-1}, then $Q$ is not uniruled by rational curves of any
degree.  If also $X$ contains a very twistable curve, then $Q$ is a
point and $X$ is isomorphic to $\PP^r$ (not merely birational to
$\PP^r$).  Thus, in what follows, $X$ will usually be of quadric type.
\end{rmk}

Let $(\zeta_1, \overline{\zeta}_e)$ be an inducting pair denoted by,
\begin{equation*}
\zeta_1 = ((p:\Sigma \rightarrow B, \sigma),g:\Sigma \rightarrow X),
\end{equation*}
\begin{equation*}
\overline{\zeta}_e = ((\overline{p}:\overline{\Sigma} \rightarrow B,
\overline{\sigma}), \overline{g}:\overline{\Sigma} \rightarrow X).
\end{equation*}
Denote
\begin{equation*}
\begin{array}{lll} 
\delta & = & \text{deg}( \zeta_1^*(2{\mc L} - {\mc H}) ), \\
\overline{\delta} & = & \text{deg}( \overline{\zeta}_e^*( \frac{2}{e}\mc{L} -
\frac{1}{e^2}\mc{H} - \sum_{i=1}^{e-1} \frac{(e-i)^2}{e^2} \Delta_i )).
\end{array}
\end{equation*}
By ~\cite[Lem. 2.2.2]{QDiv}, $\text{deg}( \sigma^*
\OO_{\Sigma}(\sigma(B))) = \delta$ and $\text{deg}(
\overline{\sigma}^* \OO_{\overline{\Sigma}}(\overline{\sigma}(B))) =
\overline{\delta}$.  In particular, both $\delta$ and
$\overline{\delta}$ are nonnegative integers, and $\overline{\delta}$
is positive by Definition ~\ref{defn-pos}.

\begin{thm} \label{thm-induction}
For each integer $d=1,\dots,\overline{\delta}$, there exists an
inducting pair $(\xi_{d,1},\overline{\xi}_{d,e+1})$ such that
$\overline{\xi}_{d,e+1}$ is typical and such that the following
conditions are satisfied,
\begin{equation} 
\label{eqn-degs}
\left\{
\begin{array}{llll}  
\text{deg}( \overline{\xi}_{d,e+1}^*\mc{H}) & = &
\text{deg}(\overline{\zeta}_e^* \mc{H}) + \text{deg}(\zeta_1^*
\mc{H}), \\
\text{deg}( \xi_{d,1}^*{\mc H} ) & = & \text{deg}(\zeta_1^* {\mc H}) + d,
& \text{ if } $X$ \text{ is planar type, } \\
\text{deg}( \xi_{d,1}^*{\mc H} ) & = & \text{deg}(\zeta_1^* {\mc H}) + 2d, 
& \text{ if } $X$ \text{ is quadric type, } \\
\text{deg} ( \xi_{d,1}^*{\mc L} ) & = & \text{deg}( \zeta_1^*{\mc L}) + d, \\
\text{deg} 
( \overline{\xi}_{d,e+1}^*{\mc L} ) & = & \text{deg}( 
\overline{\zeta}_e^*{\mc L}) + d, \\
\text{deg}( \overline{\xi}_{d,e+1}^* \Delta_{i} ) & = &
\text{deg}( \overline{\zeta}_{e}^* \Delta_{i-1} ), & \ i = 2,\dots,e-1, \\
\text{deg}( \overline{\xi}_{d,e+1}^* \Delta_{e} ) & = &
\text{deg}( \overline{\zeta}_{e}^* \Delta_{e-1} )+ \delta + d, & \text{ if } 
i> 1, \\
\text{deg}( \overline{\xi}_{d,e+1}^* \Delta_{1} ) & = & \overline{\delta}-d,
& \text{ if } i > 1, \\
\text{deg}( \overline{\xi}_{d,2}^* \Delta_{1} ) & = & \delta+\overline{\delta},
& \text{ if } i=1.
\end{array} \rt.
\end{equation}
The families are denoted,
\begin{equation*}
\overline{\xi}_{d,e+1} = (
(\overline{p}_{d,e+1}:\overline{\Sigma}_{d,e+1} \rightarrow B,
\overline{\sigma}_{d,e+1}), \overline{g}_{d,e+1}:
\overline{\Sigma}_{d,e+1} \rightarrow X), 
\end{equation*}
\begin{equation*}
\xi_{d,1} = (
(p_{d,1}: \Sigma_{d,1} \rightarrow B, \sigma_{d,1}), g_{d,1}:
\Sigma_{d,1} \rightarrow X).
\end{equation*}
Also $\overline{h}_{d,e+1}$ denotes $\overline{g}_{d,e+1} \circ
\overline{\sigma}_{d,e+1}$ and $h_{d,1}$ denotes $g_{d,1} \circ
\sigma_{d,1}$.
\end{thm}

\begin{proof}
By Lemma ~\ref{lem-posdef} (ii) and Lemma ~\ref{lem-inductable}, it
suffices to consider the case that $B$ is smooth.  By
Lemma~\ref{lem-posdef}(i) $\overline{\zeta}_e:B \rightarrow
\Kgnb{0,1}{X,e}$ is very free.  Hence, the same argument as in the
proof of Lemma~\ref{lem-posdef} (ii) and ~\cite[Prop. 3.7]{K}, it
suffices to consider the case that $\overline{\zeta}_e(B_e)$ is in
\emph{general position}, i.e., for any finite collection of
codimension $2$ subvarieties $(Z_\alpha | \alpha=1,\dots,M)$ and any
finite collection of divisors $(D_\beta | \beta=1,\dots, N)$,
$\overline{\zeta}_e(B_e)$ is disjoint from each $Z_\alpha$ and has
$0$-dimensional intersection with each $D_\beta$.

\

\textbf{The family $\zeta^{(1)}$}: Denote by $\Sigma^{(1)}$ the
surface containing $\Sigma$ and $\overline{\Sigma}$ obtained by
identifying the divisor $\sigma(B) \subset \Sigma$ and
$\overline{\sigma}(B) \subset \overline{\Sigma}$ via $\sigma(b)
\leftrightarrow \overline{\sigma}(b)$.  Denote by $p^{(1)}:
\Sigma^{(1)} \rightarrow B$ the unique morphism whose restriction to
$\Sigma$ is $p$ and whose restriction to $\overline{\Sigma}$ is
$\overline{p}$.  Denote by $g^{(1)}:\Sigma^{(1)} \rightarrow X$ the
unique morphism whose restriction to $\Sigma$ is $g$ and whose
restriction to $\overline{\Sigma}$ is $\overline{g}$.  Then
$\zeta^{(1)} = (p^{(1)}:\Sigma^{(1)} \rightarrow B, g^{(1)}:
\Sigma^{(1)} \rightarrow X)$ is a family of stable maps in the
boundary divisor $\Delta_1 \subset \Kgnb{0,0}{X,e+1}$.  Denote by
$\tau$ the stable $A$-graph with two vertices $v$ and $\overline{v}$
of degree $1$ and $e$ respectively, with no tails, and with one edge
connecting $v$ and $\overline{v}$.  Then $\zeta^{(1)}$ factors through
the canonical $1$-morphism of Behrend-Manin stacks $\Kbm(X,\tau)
\rightarrow \Kgnb{0,0}{X,e+1}$.

\begin{claim} \label{claim-1}
The pullback $(\zeta^{(1)})^* T_{\Kgnb{0,0}{X,e}}$ is ample.
\end{claim}
  
By Lemma~\ref{lem-glue1}, there is a short exact sequence,
\begin{equation}
\begin{CD}
0 @>>> \zeta_1^* T_{\text{ev}} @>>> (\zeta^{(1)})^* T_{\Kbm(X,\tau)} @>>>
\overline{\zeta}_e^* T_{\Kgnb{0,1}{X,e}} @>>> 0.
\end{CD}
\end{equation}
Since $\zeta_1$ is very twisting, by Definition ~\ref{defn-twisting}
$\zeta_1^* T_{\text{ev}}$ is ample.  Since $\overline{\zeta}_e$ is
very positive, by Lemma ~\ref{lem-posdef}, $\overline{\zeta}_e^*
T_{\Kgnb{0,1}{X,e}}$ is ample.  Hence $(\zeta^{(1)})^*
T_{\Kbm(X,\tau)}$ is ample.  By Lemma~\ref{lem-glue2}, there is a
short exact sequence,
\begin{equation}
0 \rightarrow (\zeta^{(1)})^* T_{\Kbm(X,\tau)} \rightarrow (\zeta^{(1)})^*
T_{\Kgnb{0,0}{X,e+1}} \rightarrow \sigma^* \OO_{\Sigma}(\sigma) \otimes
\overline{\sigma}^* \OO_{\overline{\Sigma}}(\overline{\sigma}) \rightarrow 0.
\end{equation}
By Definition ~\ref{defn-twisting} and Definition ~\ref{defn-pos},
both $\sigma^*\OO_{\Sigma}(\sigma)$ and $\overline{\sigma}^*
\OO_{\overline{\Sigma}}(\overline{\sigma})$ are ample.  Therefore the
tensor product is ample, and $(\zeta^{(1)})^* T_{\Kgnb{0,0}{X,e+1}}$
is ample.


\

\textbf{The family $\zeta^{(2)}$}: The family $\zeta^{(1)}$ cannot be
an inductable family, because it is a family of unpointed curves
rather than $1$-pointed curves.  The next approximation to
$\xi_{d,e+1}$ ``adds'' a marked section to $\zeta^{(1)}$.  The
self-intersection of $\overline{\sigma}(B)\subset \overline{\Sigma}$
is the degree of the invertible sheaf
$\overline{\sigma}^*\OO_{\overline{\Sigma}}(\overline{\sigma}(B))$,
which is also
\begin{equation*}
\overline{\delta} = \text{deg}\lt( 2\overline{\zeta}_e^*{\mc L} -
\overline{\zeta}_e^*{\mc H} \rt).
\end{equation*}
Let $\varsigma:B \rightarrow \overline{\Sigma}$ be a section such that
$\varsigma(B) \subset \overline{\Sigma}$ is a general member of the
linear system $|\overline{\sigma}(B)|$.  Because
$\overline{\sigma}^*\OO_{\overline{\Sigma}}(\overline{\sigma}(B))$ is
generated by global sections, there exists $\varsigma$ such that
$\varsigma(B)$ has only transverse intersections with
$\overline{\sigma}(B)$.  Denote the points of intersection by
$q_1,\dots,q_{\overline{\delta}} \in \overline{\Sigma}$.  Denote by
$w:\overline{\Sigma}^{(2)}\rightarrow \overline{\Sigma}$ the blowing
up of $\overline{\Sigma}$ at the points
$q_1,\dots,q_{\overline{\delta}}$.  Let
$\overline{p}^{(2)}:\overline{\Sigma}^{(2)} \rightarrow B$ denote the
projection $\overline{p}\circ w$.  Let $\overline{g}^{(2)}$ denote
$\overline{g}\circ w$.  Let $\overline{\sigma}^{(2)}:B \rightarrow
\overline{\Sigma}^{(2)}$ and $\varsigma^{(2)}:B \rightarrow
\overline{\Sigma}^{(2)}$ denote the proper transforms of
$\overline{\sigma}$ and $\varsigma$ respectively.  Notice that
$\overline{\sigma}^{(2)}(B)$ and $\varsigma^{(2)}(B)$ are disjoint by
construction.  So the data
\begin{equation*}
\overline{\zeta}_e^{(2)} = ( (\overline{p}^{(2)}: \overline{\Sigma}^{(2)}
\rightarrow B, \overline{\sigma}^{(2)}, \varsigma^{(2)} ),
\overline{g}^{(2)}: 
\overline{\Sigma}^{(2)} \rightarrow X )
\end{equation*}
is a family of stable $2$-pointed maps, i.e., a $1$-morphism
$\overline{\zeta}_e^{(2)}: B \rightarrow \Kgnb{0,2}{X,e}$. 

\begin{claim} \label{claim-2}
The pullback $(\overline{\zeta}_e^{(2)})^* T_{\Kgnb{0,2}{X,e}}$ is
generated by global sections.
\end{claim}

By Lemmas ~\ref{lem-def1a}, ~\ref{lem-def3a} and ~\ref{lem-def3b},
there is a short exact sequence:
\begin{equation*}
\begin{CD}
0 @>>> (\varsigma^{(2)})^*
\OO_{ {\overline{\Sigma}^{(2)}} }( \varsigma^{(1)}(B) ) @>>> 
(\overline{\zeta}_e^{(2)})^* T_{\Kgnb{0,2}{X,e}} @>>> 
\overline{\zeta}_e^* T_{\Kgnb{0,1}{X,e}} @>>> 0.
\end{CD}
\end{equation*}
By construction, the self-intersection of $\varsigma^{(2)}(B) \subset
\overline{\Sigma}^{(2)}$ is $0$, i.e., $( \varsigma^{(2)} )^* \OO_{
{\overline{\Sigma}^{(2)}} } ( \varsigma^{(2)} (B) )$ is the trivial
invertible sheaf $\OO_B$.  By hypothesis $\overline{\zeta}_e^*
T_{\Kgnb{0,1}{X,e}}$ is generated by global sections.  Therefore by
Lemma~\ref{lem-gend} (iii), $(\overline{\zeta}^{(2)}_e )^*
T_{\Kgnb{0,2}{X,e}}$ is generated by global sections.


\

Denote by $\Sigma^{(2)}$ the surface containing
$\overline{\Sigma}^{(2)}$ and $\Sigma$ obtained by identifying the
divisors $\overline{\sigma}^{(2)}(B) \subset \overline{\Sigma}^{(2)}$
and $\sigma(B) \subset \Sigma$ via $\overline{\sigma}^{(2)}(b)
\leftrightarrow \sigma(b)$.  Denote by $p^{(2)}:\Sigma^{(2)}
\rightarrow B$ the unique morphism whose restriction to
$\overline{\Sigma}^{(2)}$ is $\overline{p}^{(2)}$ and whose
restriction to $\Sigma$ is $p$.  Denote by $g^{(2)}: \Sigma^{(2)}
\rightarrow X$ the unique morphism whose restriction to
$\overline{\Sigma}^{(2)}$ is $\overline{g}^{(2)}$ and whose
restriction to $\Sigma$ is $g$.  Denote by $\sigma^{(2)}:B \rightarrow
\Sigma^{(2)}$ the morphism obtained by composing $\varsigma^{(2)}$
with the inclusion $\overline{\Sigma}^{(2)} \subset \Sigma^{(2)}$.
Then
\begin{equation*}
\zeta^{(2)} = ((p^{(2)}:\Sigma^{(2)} \rightarrow
B, \sigma^{(2)}), g^{(2)}:\Sigma^{(2)} \rightarrow X)
\end{equation*}
is a family of stable maps in the boundary $\Delta_e \subset
\Kgnb{0,1}{X,e+1}$.  Denote by $\tau^{(2)}$ the stable $A$-graph with
two vertices $v$ and $\overline{v}$ of degree $1$ and $e$
respectively, a single tail attached to $\overline{v}$, and a single
edge connecting $v$ and $\overline{v}$.  Then $\zeta^{(2)}$ factors
through the canonical $1$-morphism of Behrend-Manin stacks
$\Kbm(X,\tau^{(2)}) \rightarrow \Kgnb{0,1}{X,e+1}$.

\begin{claim} \label{claim-3}
The pullback $(\zeta^{(2)})^* T_{\Kgnb{0,1}{X,e}}$ is generated by
global sections.
\end{claim}

By Lemma~\ref{lem-glue1}, there is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> \zeta_1^* T_{\text{ev}} @>>> ( \zeta^{(2)} )^*
T_{ \Kbm(X, \tau^{(2)} ) } @>>>
( \overline{\zeta}^{(2)}_e )^* T_{\Kgnb{0,2}{X,e}} @>>> 0.
\end{CD}
\end{equation*}
By Claim ~\ref{claim-2}, the third term is generated by global
sections.  Since $\zeta_1$ is very twisting, by
Definition~\ref{defn-twisting}, $\zeta_1^* T_{\text{ev}}$ is ample.
In particular it is generated by global sections.  By
Lemma~\ref{lem-gend} (iii), $(\zeta^{(2)})^* T_{\Kbm(X,\tau^{(2)})}$
is generated by global sections.  By Lemma~\ref{lem-glue2} there is a
short exact sequence,
\begin{equation*}
0 \rightarrow ( \zeta^{(2)} )^* T_{\Kbm( X,\tau^{(2)} )} 
\rightarrow ( \zeta^{(2)} )^* T_{\Kgnb{0,1}{X,e+1}} \rightarrow 
\sigma^*
\OO_{\Sigma}(\sigma(B)) \otimes_{\OO_B} ( \overline{\sigma}^{(2)} )^*
\OO_{ \overline{\Sigma}^{(2)} }( \overline{\sigma}^{(2)}(B) )
\rightarrow 0.
\end{equation*}
By construction, $\OO_{ \overline{\Sigma}^{(2)} }(
\overline{\sigma}^{(2)}(B))$ is isomorphic to $\OO_B$.  Since
$\zeta_1$ is very twisting, by Definition~\ref{defn-twisting},
$\sigma^* \OO_{\Sigma}(\sigma(B))$ is generated by global sections.
Hence the third term in the short exact sequence is generated by
global sections.  Since also the first term is generated by global
sections, by Lemma~\ref{lem-gend} (iii), $( \zeta^{(2)} )^*
T_{\Kgnb{0,1}{X,e+1}}$ is generated by global sections.

\
  
If $e>1$, the image of $\zeta^{(2)}$ intersects the divisor
$\Delta_{1}$ transversely at the images of the points $q_1,\dots,
q_{\overline{\delta}}$.  In particular, the degree of the
$\QQ$-Cartier divisor class
$(\zeta^{(2)})^*\OO_{\Kgnb{0,1}{X,e+1}}(\Delta_{1})$ is
$\overline{\delta}$, which is positive.  If $e=1$, then
$\Delta_{1}=\Delta_{e}$.  In this case $\Kbm(X,\tau^{(2)})$ is the
normalization of $\Delta_{1,1}$ in a neighborhood of $\zeta^{(2)}(B)$.
So the degree of $(\zeta^{(2)})^*\OO_{\Kgnb{0,1}{X,2}}(\Delta_{1})$ is
the sum of the degree of the pullback of the normal sheaf of
$\Kbm(X,\tau^{(2)})\rightarrow \Kgnb{0,1}{X,2}$ and the degree of the
divisor $q_1 + \dots + q_{\overline{\delta}}$, i.e. $\delta +
\overline{\delta}$.  So also in this case the degree of
$(\zeta^{(2)})^* \OO_{\Kgnb{0,1}{X,e+1}}(\Delta_{1})$ is positive.

\

Similar computations give that $\text{deg}((\zeta^{(2)})^* \Delta_i) =
\overline{\zeta}_{e}^* \Delta_{i-1}$ for $i=2,\dots,e-1$.  The curve
$\zeta^{(2)}(B)$ is contained in the divisor $\Delta_e$, which is the
image of $\Kbm(X,\tau^{(2)})$.  As noted above, by Lemma
~\ref{lem-glue2}, the morphism $\Kbm(X,\tau^{(2)}) \rightarrow
\Kgnb{0,1}{X,e+1}$ is unramified and the pullback by $\zeta^{(2)}$ of
the normal sheaf has degree $\text{deg}(\sigma^*
\OO_{\Sigma}(\sigma(B))) = \delta$.  If $e>1$, then the image of
$\zeta^{(2)}(B)$ is contained in the smooth locus of $\Delta_e$ so
that the degree of $(\zeta^{(2)})^* \OO_{\Kgnb{0,1}{X,e+1}}(\Delta_e)$
is precisely $\delta$.  If $e=1$, then each point $q_1,\dots,
q_{\overline{\delta}}$ maps to a point of $\Delta_1$ where $\Delta_1$
intersects itself transversely.  So if $e=1$, the total degree of
$(\zeta^{(2)})^* \OO_{\Kgnb{0,1}{X,2}}(\Delta_1)$ is $\delta +
\overline{\delta}$ (this is the same result from the last paragraph --
it is included for the sake of consistency).

\

It is evident that,
\begin{equation*}
\text{deg}((\zeta^{(2)})^* \mc{H}) = \text{deg}(\overline{\zeta}_e^*
\mc{H}) + \text{deg}(\zeta_1^* \mc{H}),
\end{equation*}
and $\text{deg}((\zeta^{(2)})^* \mc{L}) =
\text{deg}(\overline{\zeta}_e^* \mc{L})$.  As a ``consistency check''
for the computations of these degrees, observe that by
~\cite[Lem. 2.2.2]{QDiv},
\begin{equation*}
\text{deg}((\sigma^{(2)})^* \OO_{\Sigma^{(2)}}(\sigma^{(2)}(B)) =
 \text{deg}\lt( (\zeta^{(2)})^* \lt( \frac{2}{e+1}\mc{L} -
\frac{1}{(e+1)^2}\mc{H} - \sum_{i=1}^e \frac{(e+1-i)^2}{(e+1)^2}
 \Delta_i \rt) \rt).
\end{equation*}
Substituting the values above, using the formulas for $\delta$ and
$\overline{\delta}$, and using that $\zeta_1^* \mc{L} =
\overline{\zeta}_{e+1}^* \mc{L}$, the degree of the right-hand-side is
$0$.  Of course this is correct; $\sigma^{(2)}(B) \subset
\Sigma^{(2)}$ has self-intersection $0$ by construction.

\

\textbf{The family $\overline{\xi}_{0,e+1}$}: Before proceeding to the
construction of the families $\overline{\xi}_{d,e+1}$, the family
$\zeta^{(2)}$ is deformed to a typical family
$\overline{\xi}_{0,e+1}$, cf. Definition ~\ref{defn-typical}.  The
families $\overline{\xi}_{d,e+1}$ will be constructed as suitable
modifications of $\overline{\xi}_{0,e+1}$.

\

Since $\zeta^{(2)}:B \rightarrow \Kgnb{0,1}{X,e+1}$ is free, i.e.,
$(\zeta^{(2)})^* T_{\Kgnb{0,1}{X,e+1}}$ is generated by global
sections, it follows by the same argument as in the proof of
Lemma~\ref{lem-posdef} (ii) and ~\cite[Prop. 3.7]{K} that there exists
a deformation $\overline{xi}_{0,e+1}:B \rightarrow \Kgnb{0,1}{X,e+1}$
of $\widetilde{\zeta}$ that is in general position.  Here
\emph{deformation} means that both $\zeta^{(2)}$ and
$\overline{\xi}_{0,e+1}$ have a common generization.  Moreover, for
any finite collection of open conditions that are satisfied by
$\zeta^{(2)}$, the deformation $\overline{\xi}_{0,e+1}$ may be chosen
to also satisfy these conditions.  Denote the family of stable maps
by,
\begin{equation*}
\overline{\xi}_{0,e+1} =
((\overline{p}_{0,e+1}:\overline{\Sigma}_{0,e+1} \rightarrow B,
\overline{\sigma}_{0,e+1}),
\overline{g}_{0,e+1}:\overline{\Sigma}_{0,e+1} \rightarrow X).
\end{equation*}
Denote by $\overline{h}_{0,e+1}:B \rightarrow X$ the composition
$\overline{g}_{0,e+1} \circ \overline{\sigma}_{0,e+1}$.

\begin{claim} \label{claim-4}
There exists a deformation $\overline{\xi}_{0,e+1}$ of $\zeta^{(2)}$
such that
\begin{enumerate}

\item[(i)]
$\overline{\xi}_{0,e+1}$ is typical, 

\item[(ii)]
$(\text{pr} \circ \overline{\xi}_{0,e+1})^* T_{\Kgnb{0,0}{X,e+1}}$ is
ample,

\item[(iii)]
$\overline{h}_{0,e+1}:B \rightarrow X$ is very twistable,

\item[(iv)]
and $(\overline{\sigma}_{0,e+1})^* \OO_{\overline{\Sigma}_{0,e+1}} (
\overline{\sigma}_{0,e+1}(B) )$ is isomorphic to $\OO_B$.

\end{enumerate}
\end{claim}

It suffices to prove that for a general deformation
$\overline{\xi}_{0,e+1}$ of $\zeta^{(2)}$, $\overline{\xi}_{0,e+1}$
satisfies the properties in Claim~\ref{claim-4}, i.e., for every
irreducible component of the Hom stack $\text{Hom}(B,\Kgnb{0,1}{X,e})$
that contains $\zeta^{(2)}$, each of the conditions (i)--(iv) holds on
a dense open substack.

\textbf{(i)}: By Lemmas ~\ref{lem-typical1} and ~\ref{lem-typical2}, a
general deformation of $\zeta^{(2)}$ is typical.

\textbf{(ii)}: The pullback by $\text{pr} \circ \zeta^{(2)}: B
\rightarrow \Kgnb{0,0}{X,e+1}$ of $T_{\Kgnb{0,0}{X,e+1}}$ equals
$(\zeta^{(1)})^* T_{\Kgnb{0,0}{X,e}}$.  By Claim~\ref{claim-1}, this
is ample.  The condition on deformations of $\zeta^{(2)}$ that the
pullback of $T_{\Kgnb{0,0}{X,e}}$ is ample is an open condition by
Lemma~\ref{lem-smDA} and Lemma~\ref{lem-opDA} (or more direct
arguments).  Therefore the pullback $(\text{pr} \circ
\overline{\xi}_{0,e+1})^* T_{\Kgnb{0,0}{X,e}}$ is ample.

\textbf{(iii)}: The morphism $g^{(2)} \circ \sigma^{(2)}:B \rightarrow
X$ equals $\overline{g} \circ \varsigma$, and this is a deformation of
$\overline{g} \circ \overline{\sigma} = g \circ \sigma$.  Because
$\zeta_1$ is very twisting, $g \circ \sigma$ is very twistable.  By
Proposition ~\ref{prop-twistopen}, $\varsigma$ can be chosen so that
$g^{(2)} \circ \sigma^{(2)}$ is very twistable.  Since
$\overline{h}_{0,e+1}$ is a deformation of $g^{(2)} \circ
\sigma^{(2)}$, by Proposition~\ref{prop-twistopen},
$\overline{\xi}_{0,e+1}$ can be chosen so that $\overline{h}_{0,e+1}$
is very twistable.

\

\textbf{(iv)}: Since $(\sigma^{(2)})^*
\OO_{\Sigma^{(2)}}(\sigma^{(2)}(B))$ is isomorphic to $\OO_B$, also
$(\overline{\sigma}_{0,e+1})^* \OO_{\overline{\Sigma}_{0,e+1}} (
\overline{\sigma}_{0,e+1}(B) )$ is isomorphic to $\OO_B$.  This proves
Claim~\ref{claim-4}.  In particular, $\overline{\xi}_{0,e+1}$ is
positive and $\overline{\xi}_{0,e+1}(B)$ is contained in the locus of
very stable maps.  Unfortunately it is \emph{not} very positive!

\

To be a bit more precise in the proof of (iii), there exists a
deformation of $\zeta_1$, $\xi_{0,1}: B \rightarrow \Kgnb{0,1}{X,1}$
that is very twisting and such that $h_{0,1}:= g_{0,1} \circ
\sigma_{0,1}:B \rightarrow X$ equals $\overline{h}_{0,e+1}$.

\

Because $\overline{\xi}_{0,e+1}$ is a deformation of $\zeta^{(2)}$,
the intersection number of $\overline{\xi}_{0,e+1}(B)$ with any
divisor of $\Kgnb{0,1}{X,e+1}$ is equal to the intersection number of
$\zeta^{(2)}(B)$ with that divisor.  These intersection numbers were
computed above.  Similarly, the intersection number of $\xi_{0,1}(B)$
with any divisor in $\Kgnb{0,1}{X,1}$ is equal to the intersection
number of $\zeta_1(B)$ with that divisor.

\

\textbf{The families $\overline{\xi}_{d,e+1}$}: The families
$\overline{\xi}_{d,e+1}$ and $\xi_{d,1}$ are constructed by induction
on $d$.  The base case $d=0$ is the pair $(\xi_{0,1},
\overline{\xi}_{0,e+1})$ constructed above, and the induction step
consists of performing a single modification and then deforming.

\

Let $d$ be an integer $1\leq d \leq \overline{\delta}$.  By way of
induction, suppose that a pair
$(\xi_{d-1,1},\overline{\xi}_{d-1,e+1})$ has been constructed such
that $\xi_{d-1,1}:B \rightarrow \Kgnb{0,1}{X,1}$ is very twisting,
$\overline{\xi}_{d-1,e+1}:B \rightarrow \Kgnb{0,1}{X,e+1}$ is very
positive and typical (in particular $\overline{\xi}_{d-1,e+1}(B)$ is
contained in the locus of very stable maps), the map $h_{d-1,1}:B
\rightarrow X$ equals the map $\overline{h}_{d-1,e+1}:B \rightarrow
X$, and the degree conditions of Equation~(\ref{eqn-degs}) hold for
$d-1$.  In particular, because $d\leq \overline{\delta}$,
$\text{deg}(\overline{\xi}_{d-1,e+1}^* \Delta_1) \geq 1$.

\

Let $b\in B$ be a closed point such that the stable map
$(\overline{\xi}_{d-1,e+1})_b$ is in $\Delta_1$.  By
Definition~\ref{defn-typical}, this stable map is of the form,
\begin{equation*}
(\overline{\xi}_{d-1,e+1})_b = ((L \cup M,
\overline{\sigma}_{d-1,e+1}(b)),(\overline{g}_{d-1,e+1})_b: L \cup M
\rightarrow X),
\end{equation*}
where $\overline{\sigma}_{d-1,e+1}(b) \in L$ and $g|_L:L \rightarrow X$ is an
isomorphism to a twistable line.  

\

Denote by $\widetilde{B}$ the prestable curve of genus $0$ containing
$B$ and $L$ obtained by identifying the divisor $b \in B$ with the
divisor $\overline{\sigma}_{d-1,e+1}(b) \in L$.  Denote by,
\begin{equation*}
\widetilde{\xi}_{d-1,e+1} = ((\widetilde{p}_{d-1,e+1}:
\widetilde{\Sigma}_{d-1,e+1} \rightarrow \widetilde{B},
\widetilde{\sigma}_{d-1,e+1}), \widetilde{g}_{d-1,e+1}:\widetilde{B}
\rightarrow X),
\end{equation*}
the modification of $\overline{\xi}_{d-1,e+1}$ associated to
$(b,L,\overline{\sigma}_{d-1,e+1}(b))$,
cf. Definition~\ref{defn-modif2}.

\

By construction, for each divisor in $\Kgnb{0,1}{X,e+1}$, the
difference of the intersection number with
$\widetilde{\xi}_{d-1,e+1}(\widetilde{B})$ and the intersection number
with $\overline{\xi}_{d-1,e+1}(B)$ equals the intersection number with
$\widetilde{\xi}_{d-1,e+1}(L)$, i.e.,
\begin{equation*}
\lt\{
\begin{array}{llll}
\widetilde{\xi}_{d-1,e+1}^* \mc{H} & = & \overline{\xi}_{d-1,e+1}^*
\mc{H}, \\
\widetilde{\xi}_{d-1,e+1}^* \mc{L} & = & \overline{\xi}_{d-1,e+1}^*
\mc{L} + 1, \\
\widetilde{\xi}_{d-1,e+1}^* \Delta_i & = & \overline{\xi}_{d-1,e+1}^*
\Delta_i, & i = 2,\dots, e-1, \\
\widetilde{\xi}_{d-1,e+1}^* \Delta_1 & = & \overline{\xi}_{d-1,e+1}^*
\Delta_1 - 1, & i > 1, \\
\widetilde{\xi}_{d-1,e+1}^* \Delta_e & = & \overline{\xi}_{d-1,e+1}^*
\Delta_e + 1, & i > 1, \\
\widetilde{\xi}_{d-1,2}^* \Delta_1 & = & \overline{\xi}_{d-1,e+1}^*
\Delta_1, & i = 1.
\end{array} \rt.
\end{equation*}
Just a few remarks about this list: The second line arises because
$\widetilde{h}_{d-1,e+1}:L \rightarrow X$ equals $g|_L:L \rightarrow
X$, which is an isomorphism to a line.  The fourth line arises by
applying Lemma~\ref{lem-glue2} and using the fact that the pullback of
$\OO_{\Kgnb{0,1}{X,e+1}}(\Delta_1)$ equals
\begin{equation*}
\sigma_{I,\lambda}^*\OO_{\Lambda}(\sigma_{I,\lambda}(L)) \otimes
(\sigma'_{I,\lambda})^* \OO_{L \times M}(\sigma'_{I,\lambda}(L)),
\end{equation*}
in the notation of Subsection~\ref{subsec-mod}, where $\lambda$ is the
point in $L\cap M$.  Of course the second factor in the tensor product
is isomorphic to $\OO_L$ and the first factor is $\OO_L(-\lambda)$
(because of the blowing up of $L \times L$ at $(\lambda,\lambda)$);
hence the net degree change is $-1$.  Similarly, the fifth line arises
because the intersection of $\widetilde{\xi}_{d-1,e+1}(L)$ and
$\Delta_e$ is exactly $\widetilde{\xi}_{d-1,e+1}(\lambda)$; hence the
degree increases by $1$.  In case $e=1$, the last two contributions
exactly cancel each other, which gives the last line.

\

As a ``consistency check'', substituting the computations above into
the formula ~\cite[Lem 2.2.2]{QDiv} for the self-intersection of
$\widetilde{\sigma}_{d-1,e+1}(\widetilde{B}) \subset
\widetilde{\Sigma}_{d-1,e+1}$ yields a net change of $+1$.  Of course
this is correct because the change equals the self-intersection of
$\sigma_{I,\Delta}(L) \subset \Lambda$, which is $T_L(-\lambda)$
(because of the blowing up of $L\times L$ at $(\lambda,\lambda)$), and
this has degree $2-1=1$.

\begin{claim} \label{claim-5}
The $1$-morphism $\widetilde{\xi}_{d-1,e+1}:\widetilde{B} \rightarrow
\Kgnb{0,1}{X,e+1}$ is very positive, the image is contained in the
locus of stable maps with only the trivial automorphism, and the
hypotheses of Lemma~\ref{lem-modif3} are satisfied.
\end{claim}

First of all, the restriction of
$\widetilde{h}_{d-1,e+1}:\widetilde{B} \rightarrow X$ to $B$ is
$\overline{h}_{d-1,e+1}$ and the restriction to $L$ is $g|_L:L
\rightarrow X$.  By hypothesis, $\overline{h}_{d-1,e+1}$ is stable,
and $g|_L$ is stable because it is a closed immersion.  Therefore
$\overline{h}_{d-1,e+1}$ is stable.  By hypothesis, the dimension of
the obstruction group of $\overline{h}_{d-1,e+1}$ is $0$.  Moreover,
possibly after deforming $\overline{\xi}_{d-1,e+1}$, it may be assumed
that the node of $L \cap M$ maps to a very general point of $X$.
Therefore $g|_L:L \rightarrow X$ is free by ~\cite[Prop. 4.14]{De}.
So the dimension of the obstruction group of $g|_L$ is $0$.  Also, by
Hypothesis ~\ref{hyp-2} it may be assumed that $\text{ev}:
\Kgnb{0,1}{X,1} \rightarrow X$ is smooth at
$((L,\overline{\sigma}_{d-1,e+1}(b)),g|_L:L \rightarrow X)$.  So by
Lemma~\ref{lem-glue1}, the dimension of the obstruction group of
$\widetilde{h}_{d-1,e+1}$ is $0$.  Hence (i) and (ii) of Definition
~\ref{defn-pos} are satisfied.

\

By hypothesis, the dimension of the obstruction group of
$\Kgnb{0,0}{X,e+1}$ at each point of
$\text{pr}(\widetilde{\xi}_{d-1,e+1}(B)) =
\text{pr}(\overline{\xi}_{d-1,e+1}(B))$ is $0$.  And
$\text{pr}(\widetilde{\xi}_{d-1,e+1}(L))$ is the point
$\text{pr}((\widetilde{\xi}_{d-1,e+1})_b)$, which is one of the points
above.  Thus (iii) of Definition~\ref{defn-pos} is satisfied.
Similarly, the restriction to $B$ of the pullback,
\begin{equation*}
(\text{pr} \circ \widetilde{\xi}_{d-1,e+1})^* T_{\Kgnb{0,0}{X,e+1}},
\end{equation*}
equals the pullback associated to $\text{pr}\circ
\overline{\xi}_{d-1,e+1}$, which is deformation ample by hypothesis.
And the restriction to $L$ is the pullback of a vector bundle by a
constant map, hence it is isomorphic to a direct sum of copies of
$\OO_L$.  In particular the restriction to $L$ is generated by global
sections.  By Lemma~\ref{lem-DAcrit}, the pullback is deformation
ample on all of $\widetilde{B}$, hence (iv) of
Definition~\ref{defn-pos} is satisfied.

\

Finally, the restriction to $B$ of the pullback,
\begin{equation*}
\widetilde{\sigma}_{d-1,e+1}^*
\OO_{ \widetilde{\Sigma}_{d-1,e+1} }( \widetilde{\sigma}_{d-1,e+1} 
( \widetilde{B} ) ),  
\end{equation*}
is equal to the analogous sheaf for $\overline{\xi}_{d-1,e+1}$, and
this is generated by global sections by construction.  The restriction
to $L$ is the pullback by the diagonal of $T_L$, twisted down by
$\overline{\sigma}_{d-1,e+1}(b)$, i.e., $\OO_L(2-1) = \OO_L(1)$.  Thus
by Lemma~\ref{lem-DAcrit}, the restriction to all of $\widetilde{B}$
is deformation ample.  Hence (v) of Definition~\ref{defn-pos} is
satisfied, therefore $\widetilde{\xi}_{d-1,e+1}$ is very positive.

\

By hypothesis $\widetilde{\xi}_{d-1,e+1}(B) =
\overline{\xi}_{d-1,e+1}(B)$ is contained in the locus of stable maps
that have only the trivial automorphism.  In particular,
$(\overline{\xi}_{d-1,e+1})_b$ has only the trivial automorphism, from
which it easily follows that $\widetilde{\xi}_{d-1,e+1}(L)$ is
contained in the locus of stable maps with only the trivial
automorphism.  Therefore the image of all of $\widetilde{B}$ is
contained in the locus of stable maps with only the trivial
automorphism.

\

Because $\overline{\xi}_{d-1,e+1}:B \rightarrow \Kgnb{0,1}{X,e+1}$ is
positive and the image is contained in the locus of very stable
curves, the hypotheses of Lemma~\ref{lem-def3b} are satisfied and
there is a short exact sequence of locally free $\OO_B$-modules,
\begin{equation*}
\begin{array}{lllllll} 
0 & \rightarrow & \overline{\sigma}_{d-1,e+1}^*
\OO_{\overline{\Sigma}_{d-1,e+1}}(\overline{\sigma}_{d-1,e+1}(B)) &
\rightarrow & \dots \\
& & \overline{\xi}_{d-1,e+1}^* T_{\Kgnb{0,1}{X,e+1}} & \rightarrow &
(\text{pr} \circ \overline{\xi}_{d-1,e+1})^* T_{\Kgnb{0,0}{X,e+1}}
& \rightarrow & 0.
\end{array}
\end{equation*}
By hypothesis, the third term is deformation ample, and the first term
is generated by global sections.  By Lemma~\ref{lem-gend} (iii), the
pullback of $T_{\Kgnb{0,1}{X,e+1}}$ is generated by global sections.
Because the node of $L\cap M$ is mapped to a very general point of
$X$, by ~\cite[Prop. 4.14]{De}, the pullback of $T_X$ to $L$ and the
pullback to $M$ are each generated by global sections; in particular
the dimension of the obstruction group of $g|_L:L \rightarrow X$ is
$0$.  Hence the hypotheses of Lemma~\ref{lem-modif3} are satisfied;
therefore Claim~\ref{claim-5} is true.

\

Of course the next step will be to apply Lemma~\ref{lem-modif3} to
construct $\overline{\xi}_{d,e+1}$.  But first the family
$\widetilde{\xi}_{d-1,1}$ is constructed.  By hypothesis, the line $L$
is twistable.  Denote by
\begin{equation*}
\xi_{L,1} = ((p_{L,1}:\Sigma_{L,1} \rightarrow L,
\sigma_{L,1}),g_{L,1}:\Sigma_{L,1} \rightarrow X),
\end{equation*}
a twisting family such that $h_{L,1} = g|_L$.  If $X$ is of planar
type, assume that the degree of $g_{L,1}$ is $1$; otherwise the degree
of $g_{L,1}$ is $2$.  By Hypotheses ~\ref{hyp-1.5} and ~\ref{hyp-2},
Lemma~\ref{lem-twisttogether2} applies to $\xi_{d-1,1}$ and
$\xi_{L,1}$, i.e., possibly after deforming the two families (without
deforming $\widetilde{h}_{d-1,e+1}$),
\begin{equation*}
(\xi_{d-1,1})_b = (\xi_{L,1})_{\overline{\sigma}_{d-1,e+1}(b)}.
\end{equation*}
Define $\widetilde{\xi}_{d-1,1}: \widetilde{B} \rightarrow
\Kgnb{0,1}{X,1}$ to be the unique $1$-morphism whose restriction to
$B$ is $\xi_{d-1,1}$ and whose restriction to $L$ is $\xi_{L,1}$.  By
Lemma~\ref{lem-twisttogether}, $\widetilde{\xi}_{d-1,1}$ is very
twisting.  By construction, $\widetilde{h}_{d-1,1} =
\widetilde{h}_{d-1,e+1}$.  Also,
\begin{equation*}
\text{deg}(\widetilde{\xi}_{d-1,1}^*\mc{H}) =
\text{deg}(\xi_{d-1,1}^*\mc{H}) + \text{deg}(g_{L,1}).
\end{equation*}
By definition, $\text{deg}(g_{L,1})$ is $1$ if $X$ is planar type, and
$2$ if $X$ is quadric type.  Finally,
\begin{equation*}
\text{deg}(\widetilde{\xi}_{d-1,1}^* \mc{L}) =
\text{deg}(\xi_{d-1,1}^* \mc{L}) + 1,
\end{equation*}
because $h_{L,1}:L \rightarrow X$ is an isomorphism to a line.

\

By Lemma~\ref{lem-modif3}, there exists a $1$-morphism
$\overline{\xi}_{d,e+1}: B \rightarrow \Kgnb{0,1}{X,e+1}$, i.e., both
$\widetilde{\xi}_{d-1,e+1}$ and $\overline{\xi}_{d,e+1}$ have a common
generization.  Since the image of $\widetilde{\xi}_{d-1,e+1}$ is
contained in the locus of stable maps with only the trivial
automorphism, $\overline{\xi}_{d,e+1}$ can be chosen with the same
property.  Because the dimension of the obstruction group at every
point of the image of $\widetilde{\xi}_{d-1,e+1}$ equals $0$, the same
is true for $\overline{\xi}_{d,e+1}$.  Hence the image of
$\overline{\xi}_{d,e+1}$ is contained in the locus of
$\Kgnb{0,1}{X,e+1}$ that is a smooth scheme.  Because
$\widetilde{\xi}_{d-1,e+1}$ can be chosen to contain a very general
point of $\Kgnb{0,1}{X,e+1}$ (because $\overline{\xi}_{d-1,e+1}$ is
typical), the same is true of $\overline{\xi}_{d,e+1}$.  By
~\cite[Prop. 4.14]{De}, the pullback $\overline{\xi}_{d,e+1}^*
T_{\Kgnb{0,1}{X,e+1}}$ is generated by global sections. Therefore,
after deforming further, $\overline{\xi}_{d,e+1}$ can be chosen to be
typical.  Because $\widetilde{\xi}_{d-1,e+1}$ is very positive, by
Lemma~\ref{lem-posopen}, $\overline{\xi}_{d,e+1}$ can be chosen to be
very positive.  Also, by the proof of Proposition
~\ref{prop-twistopen}, $\overline{\xi}_{d,e+1}$ can be chosen so that
there exists a deformation $\xi_{d,1}$ of $\widetilde{\xi}_{d-1,1}$
that is very twisting and such that $h_{d,1} = \overline{h}_{d,e+1}$.

\

Of course for any divisor in $\Kgnb{0,1}{X,e+1}$
(resp. $\Kgnb{0,1}{X,1}$), the intersection number with
$\overline{\xi}_{d,e+1}$ (resp. $\xi_{d,1}$) equals the intersection
number with $\widetilde{\xi}_{d-1,e+1}$
(resp. $\widetilde{\xi}_{d-1,1}$), and these are computed above.  This
finishes the proof of the induction step.  Hence, by induction on $d$,
for each $d=1,\dots, \overline{\delta}$, there exists an inducting
pair $(\xi_{d,1},\overline{\xi}_{d,e+1})$ as claimed.
\end{proof}

\section{Twistable lines on hypersurfaces} \label{sec-twist}

In this section it is proved that if $n+1\geq d^2$, if $X \subset
\PP^n$ is a general hypersurface of degree $d$, and if $L \subset X$
is a general line on $X$, then $L$ is twistable.

\begin{rmk} \label{rmk-exceptional}
There is one exceptional case, namely $d=1$.  For $n=1,2$, there is no
twistable line on a hyperplane in $\PP^n$, i.e., there is no twistable
line on $\PP^{n-1}$.  For $n \geq 3$ there is a twistable line on
$\PP^{n-1}$.  In this section it is proved there is a twistable line
on $\PP^{n-1}$ if $n \geq 4$.  Let $((\pi:\Sigma \rightarrow
B,\sigma),g)$ be a twisting family in $\PP^n$ such that $h:B
\rightarrow \PP^n$ is a line, and let $p\in \PP^n - g(\Sigma)$.
Denote by $\text{proj}_p:\PP^n \dashrightarrow \PP^{n-1}$ projection
from $p$.  Then $((\pi:\Sigma \rightarrow B,\sigma),\text{proj}_p
\circ g)$ is a twisting family in $\PP^{n-1}$ and $\text{proj}_p \circ
h: B \rightarrow \PP^{n-1}$ is a line.  Therefore there is also a
twistable line on $\PP^2$.
\end{rmk}

\begin{notat} \label{notat-twist}
Denote $N_d = \binom{n+d}{n}-1$ and denote by $\PP^{N_d}$ the
projective space parametrizing hypersurfaces $X\subset \PP^n$ of
degree $d$.  Denote by ${\mc X} \subset \PP^{N_d} \times \PP^n$ the
universal family of degree $d$ hypersurfaces in $\PP^n$.  Denote by
$\GG(1,n)$ the Grassmannian variety of lines in $\PP^n$.  Denote by
$F({\mc X}) \subset \PP^{N_d} \times \GG(1,n)$ the parameter space of
pairs $([X],[L])$ consisting of a hypersurface of degree $d$,
$X\subset \PP^n$, and a line $L \subset X$.  Denote by $P(t) =
(t+1)^2$ the Hilbert polynomial of a quadric surface in
$\PP^3$. Denote by $U \subset \text{Hilb}^{P(t)}_{\PP^n}$ the open
subscheme of the Hilbert scheme that parametrizes subschemes of
$\PP^n$ projectively equivalent to a smooth quadric surface in $\PP^3
\subset \PP^n$.  Denote by $V \subset U \times \GG(1,n)$ the parameter
space of pairs $([\Sigma],[L])$ consisting of a smooth quadric
surface, $\Sigma$, and a line $L \subset \Sigma$.  Denote by $V
\rightarrow \tilde{U} \rightarrow U$ the Stein factorization of the
projection $\text{pr}_U:V \rightarrow U$.  Denote by $W \subset
\PP^{N_d} \times U \times \GG(1,n)$ the parameter space of triples
$([X],[\Sigma],[L])$ consisting of a hypersurface of degree $d$,
$X\subset \PP^n$, a smooth quadric surface $\Sigma \subset X$, and a
line $L \subset \Sigma$.
\end{notat}
 
Observe that the projection $F({\mc X}) \rightarrow \GG(1,d)$ is a
projective bundle of relative dimension $N_d - (d+1)$.  Observe that
$\tilde{U} \rightarrow U$ is a finite, \'etale morphism of degree $2$.
Observe that $V \rightarrow \tilde{U}$ is a $\PP^1$-bundle.  Observe
that $W \rightarrow V$ is a projective space bundle of relative
dimension $N_d - (d+1)^2$.

\

Let $([X],[\Sigma],[L])$ be a triple in $W$.  There is a map
(well-defined up to nonzero scalar) $\partial_X:\CC^{n+1} \rightarrow
H^0(\PP^n,\OO_{\PP^n}(d-1))$ that evaluates the partial derivatives of
a defining equation of $X$.  Compose this map with the restriction map
$H^0(\PP^n,\OO_{\PP^n}(d-1)) \rightarrow
H^0(\Sigma,\OO_{\Sigma}(d-1))$, and denote the composition by,
\begin{equation*}
\partial_{X,\Sigma}:\CC^{n+1} \rightarrow H^0(\Sigma,\OO_{\Sigma}(d-1)).
\end{equation*}
Denote $E = \OO_W^{\oplus(n+1)}$.  Denote by $G$ the unique quotient of
$H^0(\PP^n,\OO_{\PP^n}(d-1)) \otimes_\CC \OO_U$ that is locally free
and whose fiber at each point $\Sigma$ is the quotient,
\begin{equation*}
H^0(\PP^n,\OO_{\PP^n}(d-1)) \rightarrow H^0(\Sigma,\OO_{\Sigma}(d-1)).
\end{equation*}
Denote by $F$ the locally free $\OO_W$-module,
\begin{equation*}
F = \text{pr}_{\PP^{N_d}}^* \OO_{\PP^{N_d}}(1) \otimes \text{pr}_U^*
G.
\end{equation*}
There is a map of $\OO_W$-modules, $\partial: E \rightarrow F$ whose
fiber at each point $([X],[\Sigma],[L])$ is the map
$\partial_{X,\Sigma}$.  Denote by $W^o \subset W$ the open subscheme
that is the complement of the support of $\text{Coker}(\partial)$,
i.e., $W^o$ is the maximal open subscheme on which $\partial$ is
surjective.

\begin{lem} \label{lem-quadric1}
Let $([X],[\Sigma],[L])$ be a point in $W^o$. Then,
\begin{enumerate}

\item[(i)] 
$X$ is smooth along $\Sigma$,

\item[(ii)]
$h^i( \Sigma, N_{\Sigma/\PP^n} ) = h^i( \Sigma, N_{\Sigma/\PP^n}(-L))
= h^i( \Sigma, N_{\Sigma/\PP^n}(-1)) = 0$, for $i>0$,

\item[(iii)]
$h^i( \Sigma, N_{\Sigma/X} )= h^i( \Sigma, N_{\Sigma/X}(-L) ) = h^i(
\Sigma, N_{\Sigma/X} \otimes \OO_{\Sigma}(-1) ) = 0$, for $i>0$,

\item[(iv)] 
$h^1(L, N_{L/X}(-1) ) = h^1( L, N_{L/X} )= 0$,

\item[(v)] 
the projection morphism $\text{pr}_{\PP^{N_d}}: W \rightarrow
\PP^{N_d}$ is smooth at $([X],[\Sigma],[L])$,

\item[(vi)] 
the projection morphism $\text{pr}_{\PP^{N_d}}: F({\mc X}) \rightarrow
\PP^{N_d}$ is smooth at $([X],[L])$, and

\item[(vii)] 
the projection morphism $\text{pr}_{F({\mc X})}:W\rightarrow F({\mc
X})$ is smooth at $([X],[\Sigma],[L])$.

\end{enumerate}
\end{lem}

\begin{proof}
\textbf{(i)}: Since the partial derivatives of a defining equation of
$X$ generate $H^0(\Sigma,\OO_{\Sigma}(d-1))$, the subscheme of $X$
where the partial derivatives all vanish is disjoint from $\Sigma$.
By the Jacobian criterion, $X$ is smooth at each point of $\Sigma$.

\

\textbf{(ii)}: Denote by $\PP^3 \subset \PP^n$ the span of $\Sigma$.
There is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> N_{\Sigma/\PP^3} @>>> N_{\Sigma/\PP^n} @>>>
N_{\PP^3/\PP^n}|_{\Sigma} @>>> 0.
\end{CD}
\end{equation*}
Since $N_{\Sigma/\PP^3} \cong \OO_{\Sigma}(2)$ and since
$N_{\PP^3/\PP^n}|_{\Sigma} \cong \OO_{\Sigma}(1)^{\oplus (n-3)}$, the
short exact sequence above is,
\begin{equation*}
\begin{CD}
0 @>>> \OO_{\Sigma}(2) @>>> N_{\Sigma/\PP^n} @>>>
\OO_{\Sigma}(1)^{\oplus (n-3)} @>>> 0.
\end{CD}
\end{equation*}
From this it is easy to compute that,
\begin{equation*}
h^i( \Sigma,N_{\Sigma/\PP^n} ) = h^i( \Sigma,N_{\Sigma/\PP^n}\otimes
\OO_{\Sigma}(-1) ) = h^i ( \Sigma,N_{\Sigma/\PP^n}\otimes
\OO_{\Sigma}(-L) ) = 0,
\end{equation*} 
for $i>0$.

\

\textbf{(iii)}: There is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> N_{\Sigma/X} @>>> N_{\Sigma/\PP^n} @>>> N_{X/\PP^n}|_\Sigma
@>>> 0.
\end{CD}
\end{equation*}
Of course $N_{X/\PP^n}|_{\Sigma} \cong \OO_{\Sigma}(d)$. For the three
cases ${\mc L} = \OO_{\Sigma}$, ${\mc L} = \OO_{\Sigma}(-L)$, and
${\mc L} = \OO_{\Sigma}(-1)$, $h^i( \Sigma, \OO_{\Sigma}(d) \otimes
{\mc L} ) = 0$ for $i>0$.  By (ii), $h^i( \Sigma, N_{\Sigma/\PP^n}
\otimes {\mc L}) = 0$ for $i>0$.  By the long exact sequence in
cohomology associated to the twist by ${\mc L}$ of the short exact
sequence above, $h^i( \Sigma, N_{\Sigma/X} \otimes {\mc L} ) = 0$ for
$i>2$.  Also $h^1(\Sigma, N_{\Sigma/X}\otimes {\mc L}) = 0$ iff the
map $e_{\mc L}: H^0(\Sigma, N_{\Sigma/\PP^n}\otimes {\mc L})
\rightarrow H^0(\Sigma, N_{X/\PP^n}|_\Sigma)$ is surjective.

\

The map $e_{\mc \OO_\Sigma}$ factors the map $\partial_{X,\Sigma}$.
Since $\partial_{X,\Sigma}$ is surjective, also $e_{\mc \OO_\Sigma}$
is surjective and $h^1(\Sigma, N_{\Sigma/X}) = 0$.

\

For the case ${\mc L} = \OO_{\Sigma}(-L)$, observe that
$\OO_{\Sigma}(1) \cong \OO_{\Sigma}(L+L')$ where $L'\subset \Sigma$ is
a line of the ruling opposite to $L$.  There is a commutative diagram,
\begin{equation*}
\begin{CD}
H^0( \Sigma , N_{\Sigma/\PP^n}(-1) ) \otimes_{\CC}
H^0( \Sigma, \OO_{\Sigma}(L') ) @>>> H^0( \Sigma, N_{\Sigma/\PP^n}(-L)) \\
@VVV @VVV \\
H^0( \Sigma , N_{X/\PP^n}|_{\Sigma}(-1) )
\otimes_{\CC} H^0( \Sigma , \OO_{\Sigma}(L') ) @>>>
H^0( \Sigma, N_{X/\PP^n}|_{\Sigma}(-L) ) \\
\end{CD}
\end{equation*}
The left vertical arrow is surjective by (ii).  The bottom arrow is
\begin{equation*}
H^0( \Sigma,
\OO_{\Sigma} ( (d-1)L + (d-1)L' ) ) \otimes H^0( \Sigma,
\OO_{\Sigma}(L') ) 
\rightarrow 
H^0( \Sigma, \OO_{\Sigma} ( (d-1)L + dL' ) ),
\end{equation*}
which is surjective.  Therefore also the right vertical arrow is also
surjective, i.e., $h^1(\Sigma, N_{\Sigma/X}(-L) )=0$.  The proof that
$h^1(\Sigma, N_{\Sigma/X})=0$ is almost identical to the proof that
$h^1(\Sigma, N_{\Sigma/X}(-L)=0$.

\

\textbf{(iv)} There is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> N_{\Sigma/X}(-1) @>>> N_{\Sigma/X}(-L') @>>> N_{\Sigma/X}|_L(-1) @>>> 0.
\end{CD}
\end{equation*}
By the associated long exact sequence in cohomology and the
computations above, $h^1\lt( L, N_{\Sigma/X}|_L(-1) \rt) = 0$.  There
is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> N_{L/\Sigma}(-1) @>>> N_{L/X}(-1) @>>> N_{\Sigma/X}|_L(-1) @>>> 0.
\end{CD}
\end{equation*}
Of course $N_{L/\Sigma} \cong \OO_L(1)$, so $h^1( L,N_{L/\Sigma}(-1)
)=0$.  As proved above, $h^1\lt( L,N_{\Sigma/X}|_L(-1) \rt) = 0$.
Therefore by the long exact sequence in cohomology, $h^1(L,
N_{L/X}(-1) ) =0$.  Since $N_{L/X}$ is a locally free $\OO_L$-module
and $h^1(L,N_{L/X}(-1))=0$, by Grothendieck's lemma $N_{L/X}$ is
generated by global sections.  In particular, also $h^1(L, N_{L/X}) =
0$.

\

\textbf{(v)}: By \cite[Prop. I.2.14.2]{K}, the obstruction space for
the relative Hilbert scheme, $\text{Hilb}^{P(t)}_{{\mc X}/\PP^{N_d}}$,
at the point $([X],[\Sigma])$ is contained in
$H^1(\Sigma,N_{\Sigma/X})$, which has dimension $0$ by (iv).  Since
the obstruction space vanishes, it follows by \cite[Thm.  2.10]{K}
that $\text{Hilb}^{P(t)}_{{\mc X}/\PP^{N_d}} \rightarrow \PP^{N_d}$ is
smooth at $([X],[\Sigma])$.  Also $W \rightarrow
\text{Hilb}^{P(t)}_{{\mc X}/\PP^{N_d}}$ is smooth.  Therefore the
composition $\text{pr}_{\PP^{N_d}}:W \rightarrow \PP^{N_d}$ is smooth
at $([X],[\Sigma],[L])$.

\

\textbf{(vi)}: The proof is almost identical to the proof of (v).

\

\textbf{(vii)}: By (v), $W \rightarrow \PP^{N_d}$ is smooth at
$([X],[\Sigma],[L])$.  By (vi), $F({\mc X}) \rightarrow \PP^{N_d}$ is
smooth at $([X],[L])$.  So the Jacobian criterion for the smoothness
of $\text{pr}_{F({\mc X})}$ at $([X],[\Sigma],[L])$ is that the map of
vertical tangent bundles, $d\text{pr}_{F({\mc X})}: T_{W/\PP^{N_d}}
\rightarrow \pi^* T_{F({\mc X})/\PP^{N_d}}$ is surjective at
$([X],[\Sigma],[L])$.  This reduces to the surjectivity of $H^0 \lt(
\Sigma, N_{\Sigma/X} \rt) \rightarrow H^0\lt( L, N_{\Sigma/X}|_L
\rt)$.  The cokernel is contained in $H^1( \Sigma, N_{\Sigma/X}(-L)
)$, which is zero by (iii).  Therefore $\text{pr}_{F({\mc X})}$ is
smooth at $([X],[\Sigma],[L])$.
\end{proof}

We associate to each $([X],[\Sigma],[L]) \in W^o$ a morphism $\zeta: L
\rightarrow \Kgnb{0,1}{X,1}$ as follows.  Let $\sigma:L \rightarrow
\Sigma$ be the inclusion and let $\text{pr}_L: \Sigma \rightarrow L$
be the unique projection such that $\sigma$ is a section of
$\text{pr}_L$.  Let $g:\Sigma \rightarrow X$ be the inclusion.  Then,
\begin{equation*}
\zeta = ( (\text{pr}_L:\Sigma \rightarrow L, \sigma),
g:\Sigma \rightarrow X ),
\end{equation*}
is a family of stable maps, i.e., a morphism $\zeta:L \rightarrow
\Kgnb{0,1}{X,1}$.  

\begin{lem} \label{lem-quadric2}
For every point $([X],[\Sigma],[L]) \in W^o$, the morphism $\zeta:L
\rightarrow \Kgnb{0,1}{X,1}$ is twisting.
\end{lem}

\begin{proof}
Since $g\circ \sigma:L \rightarrow X$ is an embedding, Axiom (i) of
Definition ~\ref{defn-twisting} is satisfied.  By
Lemma~\ref{lem-quadric1} (vi), the dimension of the obstruction group
of $\Kgnb{0,0}{X,1}$ at $[g\circ \sigma: L \rightarrow X]$ is $0$,
i.e., Axiom (ii) of Definition ~\ref{defn-twisting} is satisfied.

\

Denote by ${\mc N}$ the normal bundle of the regular embedding
$(\text{pr}_L, g): \Sigma \rightarrow L \times X$.  There is a short
exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> \text{pr}_L^* T_L @>>> {\mc N} @>>> N_{\Sigma/X} @>>> 0.
\end{CD}
\end{equation*}
By Remark~\ref{rmk-twisting} (ii), Axiom (iii) holds if $R^1
\lt(\text{pr}_L\rt)_* {\mc N}(-\sigma) = \{0\}$.  For each fiber $L'$
of $\text{pr}_L:\Sigma \rightarrow L$, ${\mc N}(-\sigma)|_{L'} \cong
N_{L'/X}(-1)$.  Since $([X],[\Sigma],[L'])$ is also in $W^o$, by
Lemma~\ref{lem-quadric1} (iv), $h^1 (L',N_{L'/X}(-1) )=0$.  Therefore
$R^1 ( \text{pr}_L )_* {\mc N}(-\sigma) = \{0\}$, i.e., Axiom (iii)
holds.

\

By Remark~\ref{rmk-twisting} (ii), $\zeta^* T_{\text{ev}} \cong (
\text{pr}_L )_* {\mc N}(-\sigma)$.  Part of the long exact sequence of
higher direct images associated to the twist by $\OO_{\Sigma}(-L)$ of
the short exact sequence above is,
\begin{equation*}
(\text{pr}_L)_* \text{pr}_L^* T_L(-\sigma(L)) \rightarrow
(\text{pr}_L)_* {\mc N}(-\sigma(L)) \rightarrow
(\text{pr}_L)_* N_{\Sigma/X}(-\sigma(L)) \rightarrow 
R^1 (\text{pr}_L)_* \text{pr}_L^* T_L(-\sigma(L))
\rightarrow 0.
\end{equation*}
For each fiber $L'$ of $\text{pr}_L$, $T_L(-\sigma(L))|_{L'} \cong
\OO_{L'}(-1)$.  Therefore $( \text{pr}_L )_* \text{pr}_L^*
T_L(-\sigma(L))=\{0\}$ and $R^1 (\text{pr}_L )_* \text{pr}_L^*
T_L(-\sigma(L))= \{0\}$, i.e., $( \text{pr}_L )_* {\mc N}(-\sigma(L))
\cong ( \text{pr}_L )_* N_{\Sigma/X}(-\sigma(L))$.

\

Let $p'\in L$ be a closed point and denote $L' =
\text{pr}_L^{-1}\lt\{p'\rt\}$.  Since $R^1\lt( \text{pr}_L \rt)_*
\text{pr}_L^* T_L(-L-L')=\{0\}$ and $R^1 \lt( \text{pr}_L \rt)_* {\mc
N}(-L -L')=\{0\}$, by the long exact sequence of higher direct images,
also $R^1 \lt( \text{pr}_L \rt)_* N_{\Sigma/X}(-L-L')=\{0\}$.  By the
Leray spectral sequence, $h^1\lt( \Sigma, N_{\Sigma/X}(-L-L') \rt) =
h^1 \lt( L, \lt( \text{pr}_L \rt)_*\lt( N_{\Sigma/X}(-\sigma(L))
\rt)(-p') \rt)$.  By Lemma~\ref{lem-quadric1} (iii), $h^1(\Sigma,
N_{\Sigma/X}(-L-L')) = 0$, so also $h^1( L,
(\text{pr}_L)_*(N_{\Sigma/X}(-\sigma(L)))(-p')) =0$.  By
Grothendieck's lemma, $(\text{pr}_L)_* N_{\Sigma/X}(-\sigma(L))$ is a
direct sum of line bundles of degree $\geq 0$, i.e., it is generated
by global sections.  Therefore Axiom (iv) is satisfied.  Finally,
$\sigma^* \OO_{\Sigma}(\sigma(L)) \cong \OO_L$, so Axiom (v) is
satisfied.
\end{proof}

\begin{prop} \label{prop-quadric}
If either,
\begin{enumerate}

\item[(i)]
$d =1$ and $n \geq 4$, or 

\item[(ii)]
$d \geq 2$ and $n+1 \geq d^2$, 

\end{enumerate}
then $\text{pr}_{F({\mc X})}:W^o\rightarrow F({\mc X})$ is dominant.
Therefore, for a general pair $([X],[L])\in F({\mc X})$, $L$ is a
twistable line on $X$.
\end{prop}

\begin{proof}
By Lemma~\ref{lem-quadric1} (v), it suffices to prove that $W^o$ is
nonempty.

\

\textbf{(i)}: If $d=1$ and $n \geq 4$, then for any quadric surface
$\Sigma \subset \PP^n$ and any hyperplane $X \subset \PP^n$ containing
$\text{span}(\Sigma)$, $\partial$ is an isomorphism so
$([X],[\Sigma],[L])$ is in $W^o$ for any line $L \subset \Sigma$.

\

\textbf{(ii)}: Next suppose that $d\geq 2$.  Denote by $I_d$ the set,
\begin{equation*} 
I_d=\{(i,j)\in \ZZ \times \ZZ : \ 0\leq i,j \leq d-1, \ i+j \geq 3\},  
\end{equation*}
which has $d^2 - 4$ elements.  Denote by,
\begin{equation*}
(Y_0,Y_1,Y_2,Y_3) \cup \lt( X_{i,j}
  \rt)_{(i,j)\in I_d} \cup (Z_m: m = 1,\dots, n+1-d^2),
\end{equation*}
a basis of $H^0(\PP^n,\OO_{\PP^n}(1))$, i.e., a basis of homogeneous
coordinates on $\PP^n$.  Denote by $\Sigma \subset \PP^n$ the smooth
quadric surface with ideal,
\begin{equation*}
I_{\Sigma} = \langle Y_0 Y_3 - Y_1 Y_2 \rangle + \langle  X_{i,j} |
(i,j) \in I_d \rangle + \langle Z_m | m= 1, \dots, n+1-d^2 \rangle.
\end{equation*}
This is the image of the closed immersion $f:\PP^1 \times \PP^1
\rightarrow \PP^n$,
\begin{equation*}
\lt( [U_0:U_1],[V_0:V_1] \rt) \mapsto [U_0V_0: U_0V_1: U_1V_0: U_1V_1:
0 : \dots :0 ].
\end{equation*}

\

For each $(i,j)\in I_d$, denote $k=\min(i,j)$, denote $i'=i-k$, and
denote $j'=j-k$.  Denote by $X\subset \PP^n$ the hypersurface with
defining equation,
\begin{equation*}
F = \lt( Y_0 Y_3 - Y_1 Y_2 \rt) Y_3^{d-2} + \sum_{(i,j) \in I_d} Y_0^k
Y_1^{i'} Y_2^{j'} Y_3^{d+k-i-j-1} X_{i,j}.
\end{equation*}
Clearly $\Sigma \subset X$.  The claim is that $\partial F:\CC^{n+1}
\rightarrow H^0\lt(\Sigma,\OO_{\Sigma}(d-1) \rt)$ is surjective.  By
construction,
\begin{equation*}
\frac{\partial F}{\partial Y_0} \mapsto U_1^{d-1}V_1^{d-1}, \
\frac{\partial F}{\partial Y_1} \mapsto U_1^{d-1}V_0V_1^{d-2}, \ 
\frac{\partial F}{\partial Y_2} \mapsto U_0U_1^{d-2}V_1^{d-1}, \
\frac{\partial F}{\partial Y_3} \mapsto U_0U_1^{d-2}V_0V_1^{d-2}.
\end{equation*}
For each $(i,j)\in I_d$,
\begin{equation}
\frac{\partial F}{\partial X_{i,j}} \mapsto
U_0^iU_1^{d-1-i}V_0^jV_1^{d-1-j}.
\end{equation}
Since the partial derivatives of the form $\frac{\partial F}{\partial
Y_i}$ give the terms $U_0^iU_1^{d-1-i}V_0^jV_1^{d-1-j}$ with $(i,j) =
(0,0), (0,1), (1,0),$ and $(1,1)$, and since these are precisely the
pairs $(i,j)$ not contained in $I_d$, $\partial F$ is surjective.
Therefore, for every line $L\subset \Sigma$, $([X],[\Sigma],[L])$ is
in $W^o$.
\end{proof}

Together with Remark ~\ref{rmk-exceptional}, Lemma~\ref{lem-quadric2}
and Proposition~\ref{prop-quadric} imply the following corollary.

\begin{cor} \label{cor-quadric}
If $X \subset \PP^n$ is a general hypersurface of degree $d$ and
either,
\begin{enumerate}

\item[(i)]
$d=1$ and $n \geq 3$, or

\item[(ii)] 
$d \geq 2$ and $n+1 \geq d^2$,

\end{enumerate} 
then Hypothesis ~\ref{hyp-2} holds.
\end{cor}

\section{Base case of the induction for hypersurfaces}
\label{sec-basecase}
In this section it is proved that if $n \geq d^2 + d + 1$ and if $X
\subset \PP^n$ is a general hypersurface of degree $d$, then there
exists a morphism $\overline{\zeta}_1:\PP^1 \rightarrow
\Kgnb{0,1}{X,1}$ that is both very twisting and very positive.  This
provides the base case for the induction argument of
Section~\ref{sec-induct}.

\begin{rmk} \label{rmk-exceptional2}
There is one exceptional case, $d=1$.  It will be proved that for $n
\geq 7$, there is a morphism $\overline{\zeta}_1: \PP^1 \rightarrow
\Kgnb{0,1}{\PP^{n-1},1}$ that is both very twisting and very positive.
As in Remark ~\ref{rmk-exceptional}, repeatedly projecting from a
point produces a morphism $\overline{\zeta}_1: \PP^1 \rightarrow
\Kgnb{0,1}{\PP^{n-1},1}$ that is very twisting and very positive for
all $n \geq 3$.
\end{rmk}

The techniques in this section are the same as those of
Section~\ref{sec-twist}.  Proposition ~\ref{prop-quadric} is proved by
finding a single degree $d$ polynomial $F$ on $\PP^n$, vanishing on
some quadric surface $\Sigma$, and such that
\begin{equation*}
\partial_{F,\Sigma}:\CC^{n+1} \rightarrow H^0\lt(\Sigma,
\OO_{\Sigma}(d-1)\rt)
\end{equation*}
is surjective.  In this section, the role of $L\subset X$ is replaced
by a rational normal curve $C_0\subset X$ of degree $k\leq n$ (in the
end, only the case $k=2d$ will be needed).  The role of the quadric
surface is replaced by a rational normal scroll $\Sigma$ of degree
$2k-1$ such that $C_0 \subset \Sigma \subset X$.  The cohomology
vanishing results of the last section are replaced by the vanishing of
$h^i( \Sigma, N_{\Sigma/X}(-C_0 - 2L) )$ for $i>0$, where $L$ is a
line of the ruling of $\Sigma$.  The computation in this section will
be to find a single degree $d$ polynomial $F$ on $\PP^n$, vanishing on
$\Sigma$, and such that the image, $W$, of the map,
\begin{equation*}
\partial_{F,\Sigma}: \CC^{n+1} \rightarrow H^0(
\Sigma,\OO_{\Sigma}(d-1) ),
\end{equation*}
has the property that the induced map,
\begin{equation*}
W \otimes H^0 (\Sigma,\OO_{\Sigma} ( (k-3)L ) ) \rightarrow
H^0 ( \Sigma, \OO_{\Sigma}(d-1) \otimes \OO_{\Sigma} ( (k-3) L ) ),
\end{equation*}
is surjective.  A similar polynomial $F$ to that of the last section
satisfies this condition.

\subsection{Generating linear systems on ${\mathbb F}_1$}
\label{subsec-glsF1}
In the last section, the relevant surface was the Hirzebruch surface
${\mathbb F}_0 = \PP^1 \times \PP^1$ embedded as a quadric surface.
In this section, the relevant surface is the Hirzebruch surface
${\mathbb F}_1$ embedded as a rational normal scroll of degree $2k-1$.
The projective model of ${\mathbb F}_1$ used here is,
\begin{equation*}
{\mathbb F}_1 = \lt\{ ([T_0:T_1],[T_0U:T_1U:V]) \in \PP^1 \times \PP^2
| T_0 (T_1U) = T_1 (T_0U) \rt\}.
\end{equation*}
In the equation above, ``$T_0U$'' and ``$T_1U$'' are just names of
homogeneous coordinates on $\PP^2$ (although the term $U$ does have a
meaning described below).  Denote by $\text{pr}_{\PP^1}:{\mathbb F}_1
\rightarrow \PP^1$ and $\text{pr}_{\PP^2}: {\mathbb F}_1 \rightarrow
\PP^2$ the projection morphisms.  Denote by $\OO_{{\mathbb F}_1}(F)$
the invertible sheaf $\text{pr}_{\PP^1}^* \OO_{\PP^1}$ and by
$\OO_{{\mathbb F}_1}(E+F)$ the invertible sheaf $\text{pr}_{\PP^2}^*
\OO_{\PP^2}$.  The invertible sheaf $\OO_{{\mathbb F}_1}(E)$ is
associated to the \emph{directrix} $E\subset {\mathbb F}_1$.  (This
explains the terminology $T_0U$ and $T_1U$; $U$ is a nonzero element
of $H^0\lt({\mathbb F}_1,\OO_{{\mathbb F}_1}(E) \rt)$, and $T_0U$ and
$T_1U$ are the products of $U$ with the two global sections $T_0$ and
$T_1$ of $H^0\lt({\mathbb F}_1, \OO_{{\mathbb F}_1}(F) \rt)$.)

\

The invertible sheaves $\OO_{{\mathbb F}_1}(E+F)$ and $\OO_{{\mathbb
F}_1}(F)$ generate the Picard group of ${\mathbb F}_1$; thus
motivating the notation,
\begin{equation*}
\OO(a,b) := \OO_{{\mathbb F}_1}\lt( a(E+F) + bF \rt).
\end{equation*}
The divisors $E+F$ and $F$ are each nef, but not ample.  Therefore
they generate the nef cone; an invertible sheaf $\OO(a,b)$ is nef iff
$0\leq a,b$, and it is ample iff $0 < a,b$.

\

Let $\OO(a,b)$ be a nef invertible sheaf, and let $W\subset H^0\lt(
{\mathbb F}_1, \OO(a,b) \rt)$ be a linear system.  Let $c \geq 0$ be
an integer.

\begin{defn} \label{defn-cgen}
The linear system $W$ is a $c$-\emph{generating linear system} if the
associated map,
\begin{equation*}
\mu_{W,c}: W \otimes H^0\lt({\mathbb F}_1, \OO(0,c)\rt)
\rightarrow H^0\lt( {\mathbb F}_1, \OO(a,b+c) \rt),
\end{equation*}
is surjective.
\end{defn}

When is $W$ a $c$-generating linear system?  

\begin{notat} \label{notat-glsF1} 
For each integer $i$, denote $r(i) = \lt\lfloor \frac{b+i-1}{c+1}
\rt\rfloor$.  Denote by $\beta_d$, $\beta_r$, $\alpha_d$ and
$\alpha_r$ the unique integers such that $0 \leq \alpha_r, \beta_r <
c+1$ and $b-1 = \beta_d(c+1) + \beta_r$, $a+b-1 = \alpha_d(c+1) +
\alpha_r$.  Denote by $N(a,b,c)$ the integer,
\begin{equation*}
N(a,b,c) = \sum_{i=0}^a (r(i) + 2) = 2a + 2 + \lt\{ \begin{array}{ll}
                    \beta_d(c+1-\beta_r) + \alpha_d(1+\alpha_r) +
                    (\alpha_d -\beta_d -1 )(c+1)  & \alpha_d >
                    \beta_d, \\
                    \beta_d(\alpha_r - \beta_r + 1) & \alpha_d =
                    \beta_d.
                 \end{array} \rt.
\end{equation*}
Denote by $W_0(a,b,c) \subset H^0(\mathbb{F}_1, \OO(a,b)
)$ the linear system,
\begin{eqnarray*}
W_0(a,b,c) = \text{span} \{ U^i V^{a-i}
T_0^{(b+i)-j(c+1)}T_1^{j(c+1)}|i=0,\dots, a,
j=0,\dots,r(i) \} \\  
+ \text{span}\lt\{U^i V^{a-i}
T_1^{b+i}\rt|i=0,\dots,a\}.
\end{eqnarray*} 
\end{notat}

\

\begin{lem} \label{lem-comput}
The linear system $W_0(a,b,c)$ is a $c$-generating linear system of
dimension $N(a,b,c)$.
\end{lem}

\begin{proof}
For each pair of nonnegative integers $a',b'$ there is a decreasing
filtration on $H^0( {\mathbb F}_1, \OO(a',b') )$,
\begin{equation*}
F^iH^0 ( {\mathbb F}_1, \OO(a',b') ) = H^0( {\mathbb F}_1,
 \OO(a',b')(-iE)  ) \cong H^0( {\mathbb F}_1, \OO(a'-i,b') ). 
\end{equation*}
For any linear system $W\subset H^0( {\mathbb F}_1, \OO(a,b))$, there
is an induced filtration $F^iW = F^i\cap W$.  The multiplication map
$\mu_{W,c}$ respects the filtrations on $W$ and on $H^0( {\mathbb
F}_1, \OO(a,b+c) )$. 
% in the strong sense that,
% \begin{equation*} \mu_{W,c}^{-1} ( F^iH^0( {\mathbb F}_1, \OO(a,b+c) ) ) =
% F^iW \otimes H^0({\mathbb F}_1, \OO(0,c)).
% \end{equation*}
Hence $\mu_{W,c}$ is surjective if every associated graded map,
\begin{equation*}
\text{gr}^i\mu_{W,c}: \text{gr}^iW \otimes H^0( {\mathbb F}_1,
\OO(0,c) ) \rightarrow \text{gr}^i H^0( {\mathbb
F}_1, \OO(a,b+c) ),
\end{equation*}
is surjective.  Now $\text{dim}(W)$ is the sum of all terms
$\text{dim}(\text{gr}^i W)$.  For
each $i$, what is the minimum possible dimension of a vector subspace $W^i
\subset \text{gr}^i H^0( {\mathbb F}_1, \OO(a,b) )$ such that the
associated map,
\begin{equation*} 
\text{gr}^i\mu_{W^i,c}: W^i \otimes H^0( {\mathbb F}_1, \OO(0,c) )
\rightarrow \text{gr}^i H^0( {\mathbb F}_1, \OO(a,b+c)),
\end{equation*}
is surjective?


\

The associated graded pieces of $\OO(a',b')$ are,
\begin{equation*}
\text{gr}^i H^0( {\mathbb F}_1, \OO(a',b') )
       \cong \lt\{ \begin{array}{ll}
                  H^0( E, \OO_E(b'+i) ), & 0\leq i \leq a' \\
                  \{0\}, & i > a' 
                \end{array} \rt. .
\end{equation*}
Let $W^i \subset H^0\lt( E, \OO_E(b+i) \rt)$ be a linear system such
that the multiplication map,
\begin{equation*}
\text{gr}^i \mu_{W^i,c}: W^i \otimes H^0 ( E, \OO_E(c) )
\rightarrow H^0 ( E, \OO_E(b+c+i) ),
\end{equation*}
is surjective.  Counting dimensions on the left and right side of the
equation, $\text{dim}(W^i)\cdot (c+1) \geq (b+c+i+1)$, i.e.,
\begin{equation*}
\text{dim}(W^i) \geq \lt\lfloor\frac{b+i-1}{c+1} \rt\rfloor +2 = r(i)
+ 2.
\end{equation*}
The linear system $W^i = \text{gr}^i
W_0(a,b,c)$, which is generated by the set of monomials
\begin{equation*}
\{ U^i V^{a-i} T_0^{(b+i)-j(c+1)}T_1^{j(c+1)}| j=0,\dots,r(i) \}\cup  
\{ U^i V^{a-i} T_1^{b+i}\} ,
\end{equation*}
has the property that $W^i \otimes H^0 ( E, \OO_E(c) ) \rightarrow 
H^0 ( E, \OO_E(b+c+i) )$ is surjective.  And $\text{dim}(W^i) =
r(i)+2$.
So $W_0(a,b,c)$ is a $c$-generating linear system of dimension,
% So the minimum dimension of a $c$-generating linear series is
% \begin{equation*}
% \text{dim}(W) = 2a+2 + \sum_{i=0}^a \lt\lfloor \frac{b+i-1}{c+1}
% \rt\rfloor,
% \end{equation*}
% and this dimension is achieved by $W_0(a,b,c)$.  By a straightforward
% computation,
\begin{equation*}
2a+2 + \sum_{i=0}^a \lt\lfloor \frac{b+i-1}{c+1} \rt\rfloor =
N(a,b,c).
\end{equation*}
Moreover, this is the minimum dimension among $c$-generating
linear systems for which each map $\text{gr}^i \mu_{W^i,c}$ is surjective.
\end{proof}

\begin{notat} \label{notat-Cox}
Denote by $S$ the Cox homogeneous coordinate ring of $\mathbb{F}_1$,
i.e.,
\begin{equation*}
S  := \oplus_{(a,b) \in \ZZ^2}
H^0\lt( {\mathbb F}_1, \OO(a,b) \rt) = \CC[T_0,T_1,U,V]. 
\end{equation*}
This is a $\ZZ^2$-graded ring, where $\text{deg}(T_0) =
\text{deg}(T_1) = (0,1)$, $\text{deg}(V) = (1,0)$ and $\text{deg}(U) =
(1,-1)$.  For every $(a,b)\in \ZZ^2$, $S_{(a,b)} = H^0\lt( {\mathbb
F}_1, \OO(a,b) \rt)$.  Denote by $\text{deg}:S \rightarrow \ZZ \cup \{
- \infty \}$ the total degree defined by $\text{deg}(M) = a+b$ for all
elements $M \in S_{(a,b)}$.  Denote by $\prec$ the graded
lexicographical monomial order on $S$ that refines the grading by
total degree by $U>V>T_0>T_1$.  For every linear system $W \subset
H^0( {\mathbb F}_1, \OO(a,b) )$ denote by $\text{IN}(W)$ the linear
system generated by the initial terms of $W$.
\end{notat}

\begin{lem}\label{lem-comput2}
If the linear system $\text{IN}(W)$ contains $W_0(a,b,c)$, then $W$ is
a $c$-generating linear system.
\end{lem}

\begin{proof}
The linear system of initial terms of $\text{Image}(\mu_{W,c})$
satisfies
\begin{equation*}
\text{IN}(W)\cdot S_{(0,c)} \subset \text{IN}\lt(\text{Image}(\mu_{W,c})
\rt).
\end{equation*}
Since $\text{IN}(W)$ contains $W_0(a,b,c)$,
\begin{equation*}
W_0(a,b,c)\cdot S_{(0,c)} \subset \text{IN}\lt( \text{Image}(\mu_{W,c})
\rt).
\end{equation*}
By Lemma~\ref{lem-comput}, $W_0(a,b,c)\cdot S_{(0,c)} = S_{(a,b+c)}$.
Hence $\text{IN}\lt( \text{Image}(\mu_{W,c}) \rt) = S_{(a,b+c)}$, and
therefore $\text{Image}(\mu_{W,c}) = S_{(a,b+c)}$.
\end{proof}

\begin{rmk} \label{rmk-impcase}
The most important case is $a=d-1$, $b=(d-1)(k-1)$ and $c=k-3$ for
positive integers $d\geq 1$ and $k\geq 3$ ($d$ will be the degree of
the hypersurface $X\subset \PP^n$, and $k$ will be the degree of the
curve $C_0\subset X$).  In particular, if $d\geq 2$ and $k\geq 2d$,
then $b-1 = (d-1)(k-2) + d-2$, $a+b-1 = (d-1)(k-2) + 2d-3$ and $0 \leq
d-2, \ 2d-3 \leq k-3$.  Hence $r(i) = d-1$ for $i=0,\dots,a$ and
$N(a,b,c) = d(d+1) = d^2 +d$.  Moreover, $\min_{k\geq 3}
N(d-1,(d-1)(k-1),(k-3)) = d^2 +d$.  This is the origin of the term
``$d^2 + d$'' in Theorem~\ref{thm-thm1}.
\end{rmk}

\subsection{Cohomology Results}

\begin{notat} \label{notat-cohres}
Let $N_d$, $\PP^{N_d}$, and $\mc{X} \subset \PP^{N_d} \times \PP^n$ be
as in Notation~\ref{notat-twist}.  Let $k$ be any integer with $3\leq
k \leq \frac{n}{2}$ (only the case $k=2d$ will be used later).  Let
${\mc R}^k(\PP^n) \subset \text{Hilb}^{kt+1}_{\PP^n}$ denote the open
subscheme parametrizing curves $C_0\subset \PP^n$ that are
projectively equivalent to a degree $k$ rational normal curve $C_0
\subset \PP^k \subset \PP^n$.  Let ${\mc R}^k({\mc X}) \subset
\PP^{N_d} \times {\mc R}^k(\PP^n)$ denote the parameter space for
pairs $([X],[C_0])$ such that $C_0\subset X$.  Let $Q(t) =
\frac{1}{2}(t+1)((2k-1)t+2)$ denote the Hilbert polynomial of a
rational normal scroll of degree $2k-1$ in $\PP^{2k}$.  Let ${\mc
U}\subset \text{Hilb}^{Q(t)}_{\PP^n}$ denote the open subscheme
parametrizing closed subschemes $\Sigma \subset \PP^n$ that are
projectively equivalent to a rational normal scroll of degree $2k-1$
in $\PP^{2k}\subset \PP^n$ and that are abstractly isomorphic to
${\mathbb F}_1$.  Let ${\mc V}\subset {\mc U} \times {\mc R}^k(\PP^n)$
denote the parameter space of pairs $([\Sigma],[C_0])$ such that
$C_0\subset \Sigma$ and such that, via the isomorphism of $\Sigma
\cong {\mathbb F}_1$, the invertible sheaf of $C_0$ is $\OO(1,0)$.
Let ${\mc W} \subset \PP^{N_d}\times {\mc U} \times {\mc R}^k(\PP^n)$
denote the parameter space for triples $([X],[\Sigma],[C_0])$ where
$([\Sigma],[C_0])$ is in ${\mc V}$ and where $\Sigma \subset X$.
\end{notat}

Observe that ${\mc R}^k\lt(\PP^n\rt)$ is a homogeneous space of
$\text{PGL}_{n+1}$, and therefore is smooth and connected.  Observe
that the projection ${\mc R}^k({\mc X}) \rightarrow {\mc R}^k(\PP^n)$
is a projective bundle of relative dimension $N_d - (kd+1)$.  Observe
that the projection map ${\mc V} \rightarrow {\mc U}$ factors as an
open subset (with nonempty fibers) of a projective bundle over ${\mc
U}$ of relative dimension $2$ (more precisely, every fiber is
isomorphic to the $\AAA^2$ of irreducible curves in the linear system
$|\OO(1,0)|$).  Observe that the projection map ${\mc W} \rightarrow
{\mc V}$ is a projective bundle of relative dimension $N_d - Q(d)$.

\

For each triple $([X],[\Sigma],[C_0]) \in {\mc W}$, define
$\partial_{X,\Sigma}: \CC^{n+1} \rightarrow H^0\lt( \Sigma,
\OO_{\Sigma}(d-1) \rt)$ as in Section~\ref{sec-twist}.  More
precisely, denote ${\mc E} = \OO_{W}^{\oplus(n+1)}$.  Denote by ${\mc
G}$ the unique quotient of $H^0(\PP^n, \OO_{\PP^n}(d-1)) \otimes_\CC
\OO_{\mc U}$ that is locally free and whose fiber at each point
$\Sigma$ is the quotient,
\begin{equation*}
H^0(\PP^n, \OO_{\PP^n}(d-1)) \rightarrow H^0\lt(\Sigma, \OO_{\Sigma}(d-1)\rt).
\end{equation*}
Denote by ${\mc F}$ the locally free $\OO_{\mc W}$-modules
\begin{equation*}
\text{pr}_{\PP^{N_d}}^*(\OO_{\PP^{N_d}}(1))\otimes \text{pr}_{\mc U}^*
{\mc G}. 
\end{equation*}
Then there is a map of $\OO_{\mc W}$-modules $\partial:{\mc E}
\rightarrow {\mc F}$ whose fiber at each point $([X],[\Sigma],[C_0])$
is the map $\partial_{X,\Sigma}$.  Denote by ${\mc W}^o\subset {\mc
W}$ the open subscheme parametrizing points $([X],[\Sigma],[C_0])$
such that,
\begin{equation*}
\text{Image}(\partial_{X,\Sigma}) \subset H^0( \Sigma,
\OO_{\PP^n}(d-3)|_\Sigma ),
\end{equation*}
is a $(k-3)$-generating linear system.

\

Let $\Sigma \subset \PP^{2k} \subset \PP^n$ be a rational normal
surface scroll of degree $2k-1$, and let $f:{\mathbb F}_1 \rightarrow
\Sigma$ be an isomorphism.  For each pair of nonnegative integers
$(a,b)$, denote by $N(a,b)$ the locally free $\OO_{{\mathbb
F}_1}$-module,
\begin{equation*}
N(a,b) = f^*\lt( N_{\Sigma/\PP^n}\otimes \OO_{\PP^n}(-1)|_\Sigma \rt) \otimes 
  \OO(a,b),
\end{equation*}
and denote by $N'(a,b) \subset N(a,b)$ the subsheaf,
\begin{equation*} 
N'(a,b) = f^*\lt( N_{\Sigma/\PP^{2k} }\otimes \OO_{\PP^n}(-1)|_\Sigma
  \rt) \otimes 
  \OO(a,b).
\end{equation*}

\begin{lem} \label{lem-scroll1}
\begin{enumerate}

\item[(i)] 
$N'(0,0)$ is generated by global sections and $h^i( {\mathbb F}_1,
N'(0,0) ) = 0$ for $i>0$,

\item[(ii)] 
$N(0,0)$ is generated by global sections and $h^i ( {\mathbb F}_1,
N(0,0) ) = 0$ for $i>0$,

\item[(iii)] 
for every pair of nonnegative integers $(a,b)$ and for every coherent
sheaf ${\mathcal F}$ on ${\mathbb F}_1$ that is generated by global
sections and such that $h^i ( {\mathbb F}_1, {\mathcal F} ) = 0$ for
$i>0$, ${\mathcal F}(a,b) := {\mathcal F}\otimes \OO(a,b)$ is
generated by global sections and $h^i ( {\mathbb F}_1, {\mathcal
F}(a,b) ) =0$ for $i>0$.

\end{enumerate}
In particular, for every pair of nonnegative integers $(a,b)$,
$N(a,b)$ (resp. $N'(a,b)$) is generated by global sections and $h^i (
{\mathbb F}_1, N(a,b) ) = 0$ for $i>0$ (resp. $h^i ( {\mathbb F}_1,
N'(a,b) ) = 0$ for $i>0$).
\end{lem}

\begin{proof}
\textbf{(i)}: The morphism $\text{pr}_{\PP^1}:{\mathbb F}_1
\rightarrow \PP^1$ is isomorphic over $\PP^1$ to projection from the
projective bundle,
\begin{equation*}
{\mathbb F}_1 \cong \PP\lt( \OO_{\PP^1}(-(k-1)) \oplus \OO_{\PP^1}(-k)
\rt).
\end{equation*}
Under this isomorphism the invertible sheaf $\OO(1,k-1)$ on $\Sigma$
corresponds to the invertible sheaf $\OO(1)$ on $\PP (
\OO_{\PP^1}(-(k-1)) \oplus \OO_{\PP^1}(-k) )$ where $\OO(1)$ is the
universal invertible quotient of $\text{pr}_{\PP^1}^*(
\OO_{\PP^1}(k-1) \oplus \OO_{\PP^1}(k))$.  Up to projective
equivalence, the morphism $f:{\mathbb F}_1 \rightarrow \PP^{2k}$ is
the closed immersion given by the complete linear system of $\OO(1)$;
in particular, $f^*\OO_{\PP^{2k}}(1) \cong \OO(1)$.  Using this
isomorphism, there is a short exact sequence of $\OO_{{\mathbb
F}_1}$-modules,
\begin{equation*}
0 \rightarrow \text{pr}_{\PP^1}^* T_{\PP^1} \rightarrow
\text{pr}_{\PP^1}^*
( \OO_{\PP^1}(1)^{\oplus (2k-1)} )\otimes f^*\OO_{\PP^{2k}}(1) \rightarrow
f^* N_{\Sigma/\PP^{2k}} \rightarrow 0.
\end{equation*}
Twisting by $f^*\OO_{\PP^{2k}}(-1)$, $N'(0,0)$ is a quotient of
$\text{pr}_{\PP^1}^* ( \OO_{\PP^1}(1)^{\oplus (2k-1)} )$.  Hence
$N'(0,0)$ is generated by global sections.  Also,
\begin{equation*}
( \text{pr}_{\PP^1} )_* ( \text{pr}_{\PP^1}^*T_{\PP^1}\otimes
f^*\OO_{\PP^{2k}}(-1) ) = \{0 \}, \ \ 
R^1 ( \text{pr}_{\PP^1} )_* (\text{pr}_{\PP^1}^*T_{\PP^1} \otimes 
f^*\OO_{\PP^{2k}}(-1) ) = \{0\}.
\end{equation*}
Twisting the short exact sequence by $f^* \OO_{\PP^{2k}}(-1)$ and
forming the associated long exact sequence of higher direct images,
$R^1 ( \text{pr}_{\PP^1} )_* (f^*N_{\Sigma/\PP^{2k}}(-1) ) = \{0\}$,
and $( \text{pr}_{\PP^1} )_* (f^*N_{\Sigma/\PP^{2k}}(-1)) \cong
\OO_{\PP^1}(1)^{\oplus (2k-1)}$.  Computing the cohomology of
$N'(0,0)$ via the Leray spectral sequence associated to
$\text{pr}_{\PP^1}:{\mathbb F}_1 \rightarrow \PP^1$, $h^i ({\mathbb
F}_1, N'(0,0) ) = 0$ for $i>0$.

\

\textbf{(ii)}: There is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> N'(0,0) @>>> N(0,0) @>>> \OO(0,0)^{\oplus (n-2k)}
@>>> 0.
\end{CD}
\end{equation*}
By (i), $h^i ( {\mathbb F}_1, \OO_{{\mathbb F}_1} ) = 0$ for
$i>0$.  Therefore $N(0,0)$ is generated by global sections and
$h^i ({\mathbb F}_1, N(0,0) )=0$ for $i>0$.

\

\textbf{(iii)}: Let ${\mathcal F}$ be a coherent sheaf on ${\mathbb
F}_1$ such that ${\mathcal F}$ is generated by global sections and
such that $h^i ( {\mathbb F}_1, {\mathcal F} )=0$ for $i>0$.  It will
be proved by double induction on $(a,b)$ that for every pair of
nonnegative integers $(a,b)$, ${\mathcal F}(a,b)$ is generated by
global sections and $h^i( \mathbb{F}_1, \mathcal{F}(a,b)) = 0$ for
$i>0$.

\

The base case is $b=0$ and is established by induction on $a$.  For
$a=0$, the result follows by hypothesis.  Let $a>0$ and, by way of
induction, suppose the result is proved for $a-1$.  Let $D\subset
{\mathbb F}_1$ be a general member of the linear system $|\OO(1,0)|$.
Then $D$ is a smooth curve isomorphic to $\PP^1$.  Since $D$ is
general, there is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> {\mathcal F}(a-1,0) @>>> {\mathcal F}(a,0) @>>> {\mathcal F}(a,0)|_D
@>>> 0.
\end{CD}
\end{equation*}
The sheaf $\mathcal{F}|_D$ is generated by global sections, and
$\OO_{{\mathbb F}_1}(a(e+f))|_D \cong \OO_{\PP^1}(a)$.  Hence also
${\mathcal F}(a,0)|_D$ is generated by global sections.  By the
induction assumption, $h^1 ( {\mathbb F}_1, {\mathcal F}(a-1,0) ) =
0$.  By the long exact sequence of cohomology associated to the short
exact sequence, every global section of ${\mathcal F}(a,0)|_D$ is the
image of a global section of ${\mathcal F}(a,0)$.  Hence ${\mathcal
F}(a,0)$ is generated by global sections.  A coherent sheaf on $\PP^1$
that is generated by global sections has no higher cohomology.
Combined with the induction assumption and the long exact sequence in
cohomology associated to the short exact sequence above, $h^i (
{\mathbb F}_1, {\mathcal F}(a,0) ) = 0$ for $i>0$.  Therefore, for
every $a>0$, ${\mathcal F}(a,0)$ is generated by global sections and
$h^i ( {\mathbb F}_1, {\mathcal F}(a,0) ) = 0$ for $i>0$.

\

Suppose that $b>0$ and suppose the result is proved for $b-1$.  Let $L
\subset {\mathbb F}_1$ be a general fiber of $\text{pr}_1$.  Then $L$
is smooth and isomorphic to $\PP^1$.  Since $L$ is general, there is a
short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> {\mathcal F}(a,b-1) @>>> {\mathcal F}(a,b) @>>> {\mathcal
  F}(a,b)|_L @>>> 0.
\end{CD}
\end{equation*}
Via the isomorphism $L \cong \PP^1$, $\OO_{{\mathbb
F}_1}(a(e+f)+bf)|_L \cong \OO_{\PP^1}(a)$.  By almost identical
arguments to those above, ${\mathcal F}(a,b)$ is generated by global
sections and $h^i ( {\mathbb F}_1, {\mathcal F}(a,b) ) = 0$ for $i>0$.
\end{proof}

Let $([X],[\Sigma],[C_0])$ be a point in $\mc{W}^o$ and let $f:
\mathbb{F}_1 \rightarrow \Sigma$ be an isomorphism.  For each pair of
nonnegative integers $(a,b)$, denote $N_X(a,b) = f^*( N_{\Sigma/X}
\otimes \OO_{\PP^n}(-1)|_\Sigma) \otimes \OO(a,b)$.

\begin{lem} \label{lem-scroll2}
\begin{enumerate}

\item[(i)] 
The hypersurface $X$ is smooth along $\Sigma$.

\item[(ii)] 
For each pair of nonnegative integers $(a,b)$, $h^i ( {\mathbb F}_1,
N_X(a,b+k-3) ) = 0$ for $i>0$.

\item[(iii)]
For every line of ruling $L\subset \Sigma$ and every nonnegative
integer $a$, $h^1 ( L, N_{L/X}(a-1) ) = 0$.

\item[(iv)] 
For every nonnegative integer $a$, $h^1 ( C_0, N_{C_0/X}(a-2) )=0$.

\item[(v)] 
The projection morphism $\text{pr}_{\PP^{N_d}}:{\mc W} \rightarrow
\PP^{N_d}$ is smooth at $([X],[\Sigma],[C_0])$.

\item[(vi)] 
For every line of ruling $L\subset \Sigma$, the projection morphism
$\text{pr}_{\PP^{N_d}}: F({\mc X}) \rightarrow \PP^{N_d}$ is smooth at
$([X],[L])$.

\item[(vii)] 
The projection morphism $\text{pr}_{\PP^{N_d}}: {\mc R}^k({\mc X})
\rightarrow \PP^{N_d}$ is smooth at $([X],[C_0])$.

\item[(viii)] 
The projection morphism $\pi: {\mc W} \rightarrow {\mc R}^k({\mc X})$
is smooth at $([X],[\Sigma],[C_0])$.

\end{enumerate}
\end{lem} 

\begin{proof}
\textbf{(i)}: Since the partial derivatives of a defining equation of
$X$ span a $c$-generating linear series, in particular they generate
the sheaf $\OO_{\Sigma}(d-1)$.  Hence, there is no point of $\Sigma$
at which all the partial derivatives vanish.  By the Jacobian
criterion, $X$ is smooth along $\Sigma$.

\

\textbf{(ii)}: There is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> N_{\Sigma/X} @>>> N_{\Sigma/\PP^n} @>>> N_{X/\PP^n}|_\Sigma
@>>> 0.
\end{CD}
\end{equation*}
Denote $\alpha = a + (d-1)$ and $\beta = b+(d-1)(k-1)$ (these are
different than $\alpha_d$, $\alpha_r$, $\beta_d$ and $\beta_r$).
There is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> N_X(a,b) @>>> N(a,b) @>>> \OO(\alpha,\beta) @>>> 0. 
\end{CD}
\end{equation*}
Since $a,b \geq 0$, by Lemma~\ref{lem-scroll1}, $h^i ( {\mathbb F}_1,
N(a,b) ) = 0$ for $i\geq 0$.  By direct computation, $h^i ( {\mathbb
F}_1,\OO(\alpha,\beta) ) = 0$ for $i \geq 0$.  Hence $h^2 ( {\mathbb
F}_1,N_X(a,b) ) = 0$, and $h^1 ( {\mathbb F}_1, N_X(a,b) ) = 0$ iff
the following map is surjective,
\begin{equation*}
H^0 ( {\mathbb F}_1, N(a,b)  ) \rightarrow H^0 ( {\mathbb F}_1,
\OO(\alpha,\beta) ).
\end{equation*}
There is a commutative diagram,
\begin{equation*}
\begin{CD}
H^0 ( {\mathbb F}_1, N(a,b)  ) \otimes H^0 ( {\mathbb F}_1,
\OO(a',b')  ) @>>> H^0 ( {\mathbb F}_1, N(a+a',b+b')  ) \\
@VVV @VVV \\
H^0 ( {\mathbb F}_1, \OO (\alpha, \beta )  ) \otimes H^0 ( {\mathbb
  F}_1, \OO(a',b')  ) @>>>
H^0 ( {\mathbb F}_1,
\OO ( \alpha+a', \beta+b'  )  )
\end{CD} .
\end{equation*}
By direct computation the bottom horizontal arrow is surjective if
$a',b' \geq 0$. Hence, if the left vertical arrow is surjective, then
also the right vertical arrow is surjective; i.e., if $h^1 ( {\mathbb
F}_1, N_X(a,b) ) = 0$ then also $h^1 ( {\mathbb F}_1, N_X(a+a',b+b') )
= 0$.  Thus (ii), is reduced to the case $a=0, b=k-3$.  In this case
the commutative diagram above factors the following commutative
diagram,
\begin{equation*}
\begin{CD}
H^0 ( {\mathbb F}_1, T_{\PP^n}  ) \otimes H^0 ( {\mathbb F}_1,
\OO(0,k-3)  ) @>>> H^0 ( {\mathbb F}_1, N(0,k-3)  ) \\
@VVV @VVV \\
H^0 ( {\mathbb F}_1, \OO ( d-1, (d-1)(k-1) ) ) \otimes H^0 ( {\mathbb
  F}_1, \OO(0,k-3) ) @>>>  
H^0 ( {\mathbb F}_1, \OO ( \alpha,\beta ) )
\end{CD} .
\end{equation*}
By definition, the composition,
\begin{equation*}
H^0 ( {\mathbb F}_1 T_{\PP^n} ) \otimes
H^0 ( {\mathbb F}_1, \OO(0,k-3) ) \rightarrow H^0 ( {\mathbb
  F}_1, \OO (\alpha, \beta ) ),
\end{equation*}
is surjective iff the triple $([X],[\Sigma],[C_0])$ is in ${\mathcal
W}^o$.  Since $([X],[\Sigma],[C_0])$ is in ${\mathcal W}^o$, the right
vertical arrow is surjective, i.e., $h^1 ( {\mathbb F}_1, N_X(0,k-3) )
= 0$.

\

\textbf{(iii)}: There is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> N_{L/\Sigma}(a-1) @>>> N_{L/X}(a-1) @>>> N_{\Sigma/X}|_L(a-1)
@>>> 0. 
\end{CD}
\end{equation*}
Since $N_{L/\Sigma} \cong \OO_L$, for all $a \geq 0$, $h^1 ( L,
N_{L/\Sigma}(a-1) ) = 0$.  Therefore it suffices to prove $h^1 ( L,
N_{\Sigma/X}|_L(a-1) ) = 0$.  Since $\OO(a-1,b)|_L \cong \OO_L(a-1)$,
there is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> N_X(a,k-3) @>>> N_X(a,k-2) @>>> N_{\Sigma/X}|_L(a-1) @>>> 0.
\end{CD}
\end{equation*}
By (ii), for $a\geq 0$ the higher cohomology of the first two terms
vanishes.  By the long exact sequence in cohomology associated to this
short exact sequence, $h^1 ( L, N_{\Sigma/X}|_L(a-1) ) = 0$ for $a\geq
0$.

\

\textbf{(iv)}: The proof is almost identical to the proof of (iii).

\

\textbf{(v)}: By \cite[Prop. 2.14.2]{K}, the obstruction space for the
relative Hilbert scheme $\text{Hilb}^{Q(t)}_{{\mc X}/\PP^{N_d}}$ at
the point $([X],[\Sigma])$ is contained in $H^1 ( \Sigma, N_{\Sigma/X}
)$.  If $([X],[\Sigma],[C_0])$ is in ${\mc W}^o$, then by (ii), $ h^1
( \Sigma, N_{\Sigma/X} ) = h^1 ( {\mathbb F}_1, N_X(1,k-1) ) = 0$.  By
\cite[Thm. 2.10]{K}, $\text{Hilb}^{Q(t)}_{{\mc X}/\PP^{N_d}}
\rightarrow \PP^{N_d}$ is smooth at $([X],[\Sigma])$.  The projection
${\mc W}^o \rightarrow \textit{Hilb}^{Q(t)}({\mc X}/\PP^{N_d})$ is an
open subset of a projective bundle, and so is smooth.  Therefore the
composite morphism ${\mc W}^o \rightarrow \PP^{N_d}$ is smooth at
$([X],[\Sigma], [C_0])$.

\

\textbf{(vi)}: The proof is very similar to the proof of (v) and uses
the vanishing, $h^1 ( L, N_{L/X} ) = 0$, which was proved in (iii).

\

\textbf{(vii)}: The proof is very similar to the proof of (v) and uses
the vanishing, $h^1 ( C_0, N_{C_0/X} ) = 0$, which was proved in (iv).

\

\textbf{(viii)}: Since ${\mc W}^o \rightarrow \PP^{N_d}$ is smooth at
$([X],[\Sigma],[C_0])$ and since ${\mc R}^k \rightarrow \PP^{N_d}$ is
smooth at $([X],[C_0])$, to prove that $\pi:{\mc W}^0 \rightarrow {\mc
R}^k({\mc X})$ is smooth at $([X],[\Sigma],[C_0])$, it suffices to
check that the derivative map $d\pi: T_{{\mc W}^0/\PP^{N_d}}
\rightarrow \pi^* T_{{\mc R}^k({\mc X})/\PP^{N_d}}$ is surjective at
$([X],[\Sigma],[C_0])$.  This reduces to the statement that $H^0 (
\Sigma, N_{\Sigma/X} ) \rightarrow H^0 ( C_0, N_{\Sigma/X}|_{C_0} )$
is surjective.  The cokernel is contained in $H^1 ( {\mathbb F}_1,
N_X(0,k-1) )$.  By (iii), $h^1( C_0, N_{\Sigma/X}|_{C_0} ) = 0$,
therefore the derivative $d\pi$ is surjective at
$([X],[\Sigma],[C_0])$.
\end{proof}

Let $([X],[\Sigma],[C_0])$ be a point of ${\mc W}$.  Denote by
$\sigma:C_0 \rightarrow \Sigma$ the inclusion and denote by
$\text{pr}_{C_0}: \Sigma \rightarrow C_0$ the unique projection
morphism such that $\sigma$ is a section of $\text{pr}_{C_0}$ (via the
isomorphism $\Sigma \cong {\mathbb F}_1$, $\text{pr}_{C_0}$
corresponds to $\text{pr}_{\PP^1}$).  Denote by $g:\Sigma \rightarrow
X$ the inclusion.  There is a family of stable maps $\zeta:C_0
\rightarrow \Kgnb{0,1}{X,1}$,
\begin{equation*}
\zeta = ((\text{pr}_{C_0}:\Sigma \rightarrow C_0, \sigma), g:\Sigma
\rightarrow X).
\end{equation*}

\begin{lem} \label{lem-scroll3}
If $([X],[\Sigma],[C_0])$ is in ${\mc W}^o$, then $\zeta:C_0
\rightarrow \Kgnb{0,1}{X,1}$ is very twisting and very positive.
\end{lem}

\begin{proof}
\textbf{Very twisting}: First the axioms of
Definition~\ref{defn-twisting} are verified.  Since $g\circ \sigma:C_0
\rightarrow X$ is an embedding, Axiom (i) of Definition
~\ref{defn-twisting} is satisfied.  By Lemma~\ref{lem-scroll2} (vii),
the dimension of the obstruction group of $\Kgnb{0,0}{X,k}$ at
$[g\circ \sigma: C_0 \rightarrow X]$ is $0$, i.e., Axiom (ii) is
satisfied.

\

The proof that Axiom (iii) holds is identical to the argument for
Axiom (iii) in the proof of Lemma~\ref{lem-quadric2}, with
Lemma~\ref{lem-quadric1} (iv) replaced by Lemma~\ref{lem-scroll2}
(iii).

\

As in the proof of Lemma~\ref{lem-quadric2}, $\zeta^*T_{\text{ev}}
\cong ( \text{pr}_{C_0} )_* N_X(0,k-1)$.  Hence $\zeta^*T_{\text{ev}}$
is ample iff $h^1 ( C_0, \zeta^*T_{\text{ev}}(-2) ) = 0$.  By a Leray
spectral sequence argument similar to the one in the proof of
Lemma~\ref{lem-quadric2}, $h^1 ( C_0, \zeta^*T_{\text{ev}}(-2) ) = h^1
( {\mathbb F}_1, N_X(0,k-3) )$, which, Lemma~\ref{lem-scroll2} (ii),
equals $0$.  Therefore $\zeta^*T_{\text{ev}}$ is an ample bundle,
i.e., Axiom (iv) is satisfied.

\

Finally, observe that $\sigma^* \OO_{\Sigma}(\sigma) \cong
\OO_{C_0}(1)$ is ample, i.e., Axiom (v) is satisfied.  Thus $\zeta$ is
a very twisting family.

\

\textbf{Very positive}: Next the axioms of Definition~\ref{defn-pos}
are verified.  Axioms (i), (ii) and (iii) follow from Axioms (i), (ii)
and (iii) of Definition~\ref{defn-twisting}, as proved above.  There
is a short exact sequence,
\begin{equation*}
\begin{CD}
0 @>>> \zeta^* T_{\text{ev}} @>>> \zeta^* T_{\Kgnb{0,1}{X,1}} @>>>
\lt( g\circ \sigma\rt)^* T_X @>>> 0.
\end{CD}
\end{equation*}
It is proved above that $\zeta^* T_{\text{ev}}$ is ample.  Moreover,
by Lemma~\ref{lem-scroll2} (iv), $N_{C_0/X}$ is ample.  Of course
$T_{C_0}$ is ample.  Therefore $T_X|_{C_0}$ is ample by
Lemma~\ref{lem-secDA} (ii).  Since the first and last term in the
short exact sequence are ample, by Lemma~\ref{lem-secDA} (ii),
$\zeta^* T_{\Kgnb{0,1}{X,1}}$ is ample.  Since $\zeta^* \text{pr}^*
T_{\Kgnb{0,0}{X,1}}$ is a quotient of $\zeta^* T_{\Kgnb{0,1}{X,1}}$,
by Lemma~\ref{lem-secDA} (i), $\zeta^* \text{pr}^*
T_{\Kgnb{0,0}{X,1}}$ is ample; i.e., Axiom (iv) is satisfied.

\

Finally, $\sigma^*\OO_{\Sigma}(\sigma) \cong \OO_{C_0}(1)$, which is
ample, i.e., Axiom (v) is satisfied.  Thus $\zeta$ is a very positive
family.
\end{proof} 

\begin{prop} \label{prop-scroll}
\begin{enumerate}

\item[(i)]
If $d \geq 3$ and $n \geq d^2 + d + 1$, then for $k=2d$, ${\mc W}^o
\rightarrow {\mc R}^k({\mc X})$ is dominant, and ${\mc R}^k({\mc X})
\rightarrow \PP^{N_d}$ is dominant.

\item[(ii)]
If $d = 1$ or $2$ and if $n \geq 7$, then for $k=3$, ${\mc W}^o
\rightarrow {\mc R}^3({\mc X})$ is dominant, and ${\mc R}^3({\mc X})
\rightarrow \PP^{N_d}$ is dominant.

\end{enumerate}
\end{prop}

\begin{proof}
By item $(8)$ of lemma~\ref{lem-scroll2}, it suffices to prove that
${\mc W}^o$ is nonempty.  We have to find a pair $([X],[\Sigma])$ such
that for $a=d-1$, $b=(d-1)(k-1)$ and for $c=k-3$, we have that the
image of the derivative map
\begin{equation}
d_{X,\Sigma}:H^0\lt( (\PP^n,T_{\PP^n}(-1) \rt) \rightarrow H^0\lt(
{\mathbb F}_1, \OO(a,b) \rt)
\end{equation}
is a $c$-generating linear system.

\

Recall that $S = \CC[T_0,T_1,U,V]$ is the $\ZZ^2$-graded Cox
homogeneous coordinate ring of ${\mathbb F}_1$.  Denote by $A_d$ the
set of $d^2 + d$ monomials that occur in the linear system
$W_0(a,b,c)$, i.e.,
\begin{equation*}
\begin{array}{lr}
A_d = &  
\lt\{ U^i V^{d-1-i}
T_0^{((d-1)(k-1)+i)-j(k-2)}T_1^{j(k-2)}|i=0,\dots, d-1,
j=1,\dots,r(i) \rt\}  \\
 & \cup \lt\{U^i V^{d-1-i}
T_1^{(d-1)(k-1)+i}\rt|i=0,\dots,d-1
\}
\end{array}
\end{equation*}
where $r(i) = d-1+\lt\lfloor\frac{d-2+i}{k-2} \rt\rfloor$.

\textbf{(i), $d \geq 4$}: Suppose that $d \geq 4$ and $n \geq d^2 + d
+1$.  Denote by $B_d$ the set of $4d-1$ monomials,
\begin{equation*}
\begin{array}{lllllllll}
B_d =  & 
 & \{ & U^{d-1} &  & T_0^{(d-1)k - (k-2)j} & T_1^{(k-2)j} & | j=0,\dots, d-1
 & \} \\
 & \cup &
\{ & U^{d-2} & V & T_0^{(d-1)k-1-(k-2)j} & T_1^{(k-2)j} & | j=0,\dots, d-1
 & \} \\
 & \cup &
\{ & U^{d-3} & V^2 & T_0^{(d-1)k-2-(k-2)j} & T_1^{(k-2)j} & |
j=1,\dots, d-1 & \} \\
 & \cup &
\{ & U^{d-4} & V^3 & T_0^{(d-1)k-3-(k-2)j} & T_1^{(k-2)j} & | j=1,\dots,
d-1 & \} \\
 & \cup & \{ & & V^{d-1} & T_0^{(d-1)(k-1)} & & & \}.
\end{array}
\end{equation*}
Denote by $C_d$ the set of $d^2 - 3d + 1$ monomials $C_d = A_d - B_d$.
Denote by
\begin{equation*}
\lt\{Y_0,\dots,Y_{k} \rt\} \cup \lt\{ Z_0, \dots, Z_{k-1} \rt\}
\cup \lt\{ X_M | M \in C_d \rt\} \cup
\lt\{V_l | l=1,\dots, n-(d^2+d+1) \rt\},
\end{equation*}
a basis of $H^0(\PP^n,\OO_{\PP^n}(1))$, i.e., a basis of homogeneous
coordinates on $\PP^n$.

\

Denote by $f:{\mathbb F}_1 \rightarrow \PP^{2k} \subset \PP^n$ the
morphism mapping $([T_0:T_1],[T_0U:T_1U:V]) \in \mathbb{F}_1$ to the
point in $\PP^n$ with coordinates $X_M = 0, M\in C_d$, with $V_l=0,
l=1,\dots, n-(d^2+d+1)$, and with
\begin{eqnarray*}
Y_0 = U T_0^k,\dots, Y_i = U T_0^{k-i}T_1^i, \dots, Y_k = UT_1^k, \\
Z_{0} = V T_0^{k-1},\dots, Z_{j} = V T_0^{k-1-j}T_1^j,\dots,
Z_{k-1} = VT_1^{k-1}.
\end{eqnarray*}
This is an embedding whose image $\Sigma = f({\mathbb F}_1)$ is a
rational normal scroll of degree $2k-1$.

\

The pullback map $H^0\lt( (\PP^{2k},\OO_{\PP^{2k}}(1) \rt) \rightarrow
H^0\lt( {\mathbb F}_1, \OO(1,k-1) \rt)$ is surjective by construction.
And the natural map,
\begin{equation*}
\text{Sym}^{d-1}H^0 ( {\mathbb F}_1, \OO(1,k-1) ) \rightarrow
H^0 ( {\mathbb F}_1, \OO (d-1,(d-1)(k-1) ) ),
\end{equation*} 
is surjective.  Therefore the pullback map
\begin{equation*}
H^0\lt( \PP^{2k},\OO_{\PP^{2k}}(d-1) \rt) \rightarrow H^0\lt( {\mathbb
F}_1, \OO\lt(d-1,(d-1)(k-1) \rt) \rt)
\end{equation*}
is surjective.  For each monomial $M\in C_d$, choose a polynomial
$G_M(Y_0,\dots,Y_{2k})$ such that $f^*G_M = M$.

\

Consider the hypersurface $X\subset \PP^n$ with defining equation
\begin{equation*}
\begin{array}{llllll}
F = & \sum_{j=0}^{d-2} & \lt(Y_jZ_{j+1} - Y_{j+1}Z_{j}\rt) &
Y_0^{d-2-j}Y_{k-3}^i & + \\
 & \sum_{j=2}^{d-2} & \lt(Y_{d-1+j}Z_{d+j} - Y_{d+j}Z_{d-1+j} \rt) &
Y_k^{j-2}Y_{3}^{d-2-l}Z_3Z_4 & + \\
 & & (Y_{d-1}Z_d - Y_d Z_{d-1}) & Y_{k-3}^{d-4}Y_{k-4}^2 & + \\
 & & (Y_d Z_{d+1} - Y_{d+1}Z_d) & Y_1^{d-4}Z_7 Z_8 & + \\
 & & (Y_{k-2}Z_{k-1} - Y_{k-1}Z_{k-2}) & Y_k^{d-4}Z_{k-1}Z_5 & + \\
 & & (Y_k Z_0 - Z_dY_d) & Z_0^{d-2} & + \\
 & \sum_{M\in C_d}& G_M(Y_0,\dots,Y_{2k})X_M.
\end{array}
\end{equation*}
Observe that $F$ is contained in the homogeneous ideal of $\Sigma$,
i.e., $\Sigma \subset X$.  The derivative map $\partial_{X,\Sigma}$
acts on the coordinates $Y_j,Z_j, X_M$ as in Figure
~\ref{figure-table}.  Each of the monomials in $A_d$ occurs as the
initial term of $\partial_{X,\Sigma}$ acting on one coordinate.  For
every coordinate except $Y_0$ and $Z_0$, the initial term of
$\partial_{X,\Sigma}$ is one of the monomials in $A_d$.  By
Lemma~\ref{lem-comput2}, the image of $\partial_{X,\Sigma}$ is a
$(k-3)$-generating linear system.

\begin{figure} 
\begin{equation*}
\begin{array}{||l|l|l||}
\hline
\text{coordinate} & \text{IN}(\partial_{X,\Sigma}(\text{coordinate})) &
\text{lower order terms in } \partial_{X,\Sigma}(\text{coordinate}) \\
 & & \\

\hline
\hline
Z_{j+1}, & U^{d-1}T_0^{(k-1)d - (k-2)j}T_1^{(k-2)j} & - U^{d-1}T_0^{(k-1)d
  - (k-2)(j+1) - 1}T_1^{(k-2)(j+1) + 1} \\
j = 0,\dots, d-1 & & \\

\hline 
Z_d & U^{d-1}T_0^{(k-1)d - (k-2)(d-1)}T_1^{(k-2)(d-1)} & -
U^{d-3}V^2T_0^{(k-1)(d-2)+(d-3)} T_1^{k-1} \\
 & & - UV^{d-2}T_0^{ (k-1)(d-2)+d} T_1^d \\

\hline
Y_{j+1}, & - U^{d-2}VT_0^{(k-1)d - 1 - (k-2)j}T_1^{(k-2)j} &
+ U^{d-2}VT_0^{(k-1)d - (k-2)(j+1) - 2}T_1^{(k-2)(j+1)+1} \\
j=0, \dots, d-1 & & \\

\hline
Y_d & - U^{d-2}V T_0^{(k-1)d - 1 - (k-2)(d-1)}T_1^{(k-2)(d-1)} &
+U^{d-4}V^2 T_0^{(k-1)(d-2)+(d-4)}T_1^{k-1} \\
 & & - V^{d-1}T_0^{(k-1)(d-1)-d}T_1^d \\

\hline
Z_{d+j},          & U^{d-3} V^2 \times & - U^{d-2} V^2\times  \\
j = 1, \dots, d-2 & T_0^{(k-1)(d-1) + (d-3) - (k-2)j }T_1^{(k-2)j }
& T_0^{ (k-1)(d-1) + (d-4) + (k-2)(j+1) }T_1^{(k-2)(j+1)}  \\

\hline
Z_{k-1} & U^{d-3} V^2 \times &  \\
        & T_0^{(k-1)(d-1) + (d-3) - (k-2)(d-1)}T_1^{(k-2)(d-1)} & \\

\hline
Y_{d+j},          & -U^{d-4}V^3 \times & U^{d-4}V^3 \times \\
j = 1, \dots, d-2 & T_0^{(k-1)(d-1) + (d-4) - (k-2)j}T_1^{(k-2)j} 
& T_0^{ (k-1)(d-1) + (d-5) + (k-2)(j+1) } T_1^{(k-2)(j+1)} \\

\hline
Y_{k-1} & U^{d-4}V^3 \times & \\
        &  T_0^{(k-1)(d-1) + (d-4) - (k-2)(d-1)}T_1^{(k-2)(d-1)} & \\

\hline
Y_k & V^{d-1} T_0^{(k-1)(d-1)} & \\

\hline
X_M, & M & \\
M \in C_d & & \\

\hline 
\hline 
Z_0 & -U^{d-1}T_0^{(k-1)(d-1)+(d-2)}T_1 &
UV^{d-2}T_0^{(k-1)(d-2)}T_1^k \\

\hline
Y_0 & U^{d-2}VT_0^{(k-1)(d-1)+(d-3)}T_1 & \\

\hline
\end{array}
\end{equation*}
\caption{The map $\partial_{X,\Sigma}$}
\label{figure-table}
\end{figure}

\

\textbf{(i), $d=3$}: Suppose that $d=3$ and $n \geq 3^2 + 3 + 1 = 13$.
Denote by $B_3$ the set of $4d-1 = 11$ monomials,
\begin{equation*}
\begin{array}{lllllllll}
B_d = & 
 & \{ & U^2 &     & T_0^{12-(k-2)j} & T_1^{(k-2)j} & |j=0,1,2 & \} \\
 & \cup
 & \{ & U   & V   & T_0^{11-(k-2)j} & T_1^{(k-2)j} & |j=0,1,2 & \} \\
 & \cup
 & \{ &     & V^2 & T_0^{10-(k-2)j} & T_1^{(k-2)j} & |j=0,1,2 & \} \\
 & \cup 
 & \{ & U   & V   &                 & T_1^{11}     &          & \} \\
 & \cup
 & \{ &     & V^2 &                 & T_1^{10}     &          & \}.
\end{array}
\end{equation*}
Denote by $C_3$ the singleton set consisting of the monomial $M =
U^2T_1^{12}$.  Denote by
\begin{equation*}
\lt\{Y_0,\dots,Y_6 \rt\} \cup \lt\{ Z_0,\dots, Z_5 \rt\} \cup \lt\{
X_M | M \in C_3 \rt\} \cup \lt\{V_l | l = 1\dots, n-13 \rt \},
\end{equation*}
a basis of $H^0(\PP^n,\OO_{\PP^n}(1))$, i.e., a basis of homogeneous
coordinates on $\PP^n$.

\

Denote by $f:{\mathbb F}_1 \rightarrow \PP^{12} \subset \PP^n$ the
morphism mapping $([T_0:T_1],[T_0U:T_1U:V]) \in \mathbb{F}_1$ to the
point in $\PP^n$ with coordinates $X_M = 0, M\in C_d$, with $V_l=0,
l=1,\dots, n-13$, and with
\begin{eqnarray*}
Y_0 = U T_0^6,\dots, Y_i = U T_0^{6-i}T_1^i, \dots, Y_k = U T_1^6, \\
Z_{0} = V T_0^{5}, \dots, Z_{j} = V T_0^{5-j}T_1^j,\dots,
Z_{5} = V T_1^{5}.
\end{eqnarray*}
This is an embedding whose image $\Sigma = f({\mathbb F}_1)$ is a
rational normal scroll of degree $11$.  

\

Consider the hypersurface $X \subset \PP^n$ with defining equation
\begin{equation*}
\begin{array}{llllll}
F = & \sum_{j=0}^{1} & \lt(Y_jZ_{j+1} - Y_{j+1}Z_{j}\rt) &
Y_0^{1-j}Y_{3}^i & + \\
 & & \lt( Y_2Z_3 - Y_3Z_2 \rt) & Y_6 & + \\
 & & \lt( Y_5Z_5 - Y_6Z_4 \rt) & Z_5 & + \\
 & & \lt( Y_3Z_3 - Y_6Z_0 \rt) & Z_4 & + \\
 & & \lt( Y_2Z_2 - Y_4Z_0 \rt) & Z_0 & + \\
 & &  Y_6^2 X_M.
\end{array}
\end{equation*}
Observe that $F$ is contained in the homogeneous ideal of $\Sigma$,
i.e., $\Sigma \subset X$.  It is straightforward to compute the action
of $\partial_{X,\Sigma}$ on the coordinates $Y_j,Z_j,X_M$.  Every
monomial in $A_3$ occurs as the initial term of $\partial_{X,\Sigma}$
acting on one coordinate.  For every coordinate except $Y_0$ and
$Z_0$, the initial term of $\partial_{X,\Sigma}$ is one of the
monomials in $A_3$.  By Lemma~\ref{lem-comput2}, the image of
$\partial_{X,\Sigma}$ is a $(k-3)$-generating linear system.

\

\textbf{(ii), $d=2$}: Suppose that $d =2$ and $n \geq 2^2 + 2 +1 = 7$.
Let $k=3$.  Denote by $B_2$ the set of $6$ monomials,
\begin{equation*}
\begin{array}{lllllllll}
B_2 = &
 & \{ & U & & T_0^{3-j} & T_1^j & | j = 0, 1,2 & \} \\
 & \cup &
\{ & & V & T_0^{2-j} & T_1^j & | j = 0,1,2 & \}.
\end{array}
\end{equation*}
Denote by $C_2$ the singleton set consisting of the monomial $M =
UT_1^3$.  
Denote by
\begin{equation*}
\lt\{ Y_0,Y_1,Y_2,Y_3,Z_0,Z_1,Z_2,X_M \rt\} \cup \lt\{V_l | l=1,\dots,
n-7 \rt\},
\end{equation*}
a basis of $H^0(\PP^n,\OO_{\PP^n}(1))$, i.e., a basis of homogeneous
coordinates on $\PP^n$.

\

Denote by $f:{\mathbb F}_1 \rightarrow \PP^{6} \subset \PP^n$ the
morphism mapping $([T_0:T_1],[T_0U:T_1U:V]) \in \mathbb{F}_1$ to the
point in $\PP^n$ with coordinates $X_M = 0, M\in C_d$, with $V_l=0,
l=1,\dots, n-7$, and with
\begin{eqnarray*}
Y_0 = U T_0^3, Y_1 = UT_0^2 T_1, Y_2 = U T_0T_1^2, Y_3 = U T_1^3, \\
Z_{0} = V T_0^{2}, Z_{1} = V T_0T_1,
Z_{2} = V T_1^{2}.
\end{eqnarray*}
This is an embedding whose image $\Sigma = f({\mathbb F}_1)$ is a
rational normal scroll of degree $6$.  

\

Consider the hypersurface $X \subset \PP^n$ with defining equation
\begin{equation*}
F = (Y_3Z_1 - Y_2Z_2) + (Y_0Z_1 - Y_1Z_0) + Y_3X_M.
\end{equation*}
Observe that $F$ is contained in the homogeneous ideal of $\Sigma$,
i.e., $\Sigma \subset X$.  It is straightforward to compute the action
of $\partial_{X,\Sigma}$ on the coordinates $Y_j,Z_j,X_M$.  Every
monomial in $A_2$ occurs as the initial term of $\partial_{X,\Sigma}$
acting on one coordinate.  For every coordinate except $Y_0$, the
initial term of $\partial_{X,\Sigma}$ is one of the monomials in
$A_2$.  By Lemma~\ref{lem-comput2}, the image of $\partial_{X,\Sigma}$
is a $(k-3)$-generating linear system.

\

\textbf{(ii), $d=1$}.  Denote by
\begin{equation*}
\lt\{ Y_0,Y_1,Y_2,Y_3,Z_0,Z_1,Z_2,X_M \rt\} \cup \lt\{V_l | l=1,\dots,
n-7 \rt\},
\end{equation*}
a basis of $H^0(\PP^n,\OO_{\PP^n}(1))$, i.e., a basis of homogeneous
coordinates on $\PP^n$.

\

Denote by $f:{\mathbb F}_1 \rightarrow \PP^{6} \subset \PP^n$ the
morphism mapping $([T_0:T_1],[T_0U:T_1U:V]) \in \mathbb{F}_1$ to the
point in $\PP^n$ with coordinates $X_M = 0, M\in C_d$, with $V_l=0,
l=1,\dots, n-7$, and with
\begin{eqnarray*}
Y_0 = U T_0^3, Y_1 = UT_0^2 T_1, Y_2 = U T_0T_1^2, Y_3 = U T_1^3, \\
Z_{0} = V T_0^{2}, Z_{1} = V T_0T_1,
Z_{2} = V T_1^{2}.
\end{eqnarray*}
This is an embedding whose image $\Sigma = f({\mathbb F}_1)$ is a
rational normal scroll of degree $6$.

\

Consider the hypersurface $X \subset \PP^n$ with defining equation $F
= X_M$.  Observe that $F$ is contained in the homogeneous ideal of
$\Sigma$, i.e., $\Sigma \subset X$.  The image of
$\partial_{X,\Sigma}(X_M)$ is a generator for $S_{0,0}$; i.e., the
image of $\partial_{X,\Sigma}$ is a $(k-3)$-generating linear system.
\end{proof}

Together with Remark ~\ref{rmk-exceptional2}, Lemma~\ref{lem-scroll3}
and Proposition ~\ref{prop-scroll} imply the following corollary.

\begin{cor} \label{cor-scroll}
If $X \subset \PP^n$ is a general hypersurface of degree $d$ and if $n
\geq d^2 + d +1$, then there exists a $1$-morphism $\zeta:C_0
\rightarrow \Kgnb{0,1}{X,1}$ that is both very twisting and very
positive.
\end{cor}

\section{Proof of the main theorem}

As explained at the end of Section~\ref{sec-results}, if $d<
\frac{n+1}{2}$, then for a general hypersurface $X\subset \PP^n$ of
degree $d$, Hypothesis~\ref{hyp-1}, Hypothesis~\ref{hyp-1.5}, and
Hypothesis~\ref{hyp-1.75} are satisfied.  By
Corollary~\ref{cor-quadric}, if $d\geq 2$ and $n+1 \geq d^2$, or if
$d=1$ and $n \geq 3$, then for a general hypersurface $X \subset
\PP^n$ of degree $d$, Hypothesis~\ref{hyp-2} is satisfied.  Finally,
if $n \geq d^2 + d +1$ then by Corollary ~\ref{cor-scroll} there
exists a very twisting, very positive family $\zeta:C_0 \rightarrow
\Kgnb{0,1}{X,1}$.  Therefore $(\zeta,\zeta)$ is an inducting pair.

\

By Theorem~\ref{thm-induction}, for every $e \geq 1$ there exists an
inducting pair $(\zeta_1,\overline{\zeta}_e)$.  In particular, there
exists a very positive $1$-morphism $\overline{\zeta}_e:C \rightarrow
\Kgnb{0,1}{X,e}$.  As shown in the proof of
Theorem~\ref{thm-induction}, it may be assumed that $C$ is smooth and
that the image of $C$ is contained in the smooth locus of the fine
moduli locus.  By Lemma~\ref{lem-posdef} (i), $\overline{\zeta}_e$ is
a very free morphism.  By \cite[Prop. 7.4]{HRS2}, $\Kgnb{0,0}{X,e}$ is
an irreducible variety.  Therefore, by \cite[Thm. IV.3.7]{K},
$\Kgnb{0,0}{X,e}$ is rationally connected.

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