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\title{Divisors on the space of maps to Grassmannians} 
\author[Coskun]{Izzet Coskun}\address{Department of Mathematics,
M.I.T., Cambridge, MA 02139}
\email{coskun@math.mit.edu} 
\author[Starr]{Jason Starr} \address{Department of Mathematics,
M.I.T., Cambridge, MA 02139}
\email{jstarr@math.mit.edu}
\subjclass[2000]{Primary:14D20, 14E99, 14H10} \thanks{During the
preparation of this article the second author was partially supported
by the NSF grant DMS-0353692 and a Sloan Research Fellowship.}



\begin{document}

\begin{abstract}
  In this note we study the divisor theory of the Kontsevich moduli
  spaces $\Kgnb{0,0}(G(k,n),d)$ of genus-zero stable maps to the
  Grassmannians.  We calculate the classes of several geometrically
  significant divisors. We prove that the cone of effective divisors
  stabilizes as $n$ increases and we determine the stable effective
  cone. We also characterize the ample cone.
\end{abstract}

\maketitle
\tableofcontents


\section{Introduction}\label{intro}


The cones of ample and effective divisors are two of the most
fundamental invariants of a variety.  They control the birational and
projective geometry of the variety. In this paper we determine the
ample and stable effective cones of Kontsevich moduli spaces
$\Kgnb{0,0}(G(k,n),d)$ of genus-zero stable maps to Grassmannians.
\smallskip

The papers \cite{coskun:nef} and \cite{coskun:effective} introduce
techniques for studying the ample and effective cones of Kontsevich
moduli spaces and determine these cones when the target is projective
space. The techniques developed in \cite{coskun:nef} and
\cite{coskun:effective} apply to more general targets, as we
illustrate here with the example of Grassmannians. Our first main
theorem is: \smallskip



\begin{theorem}\label{amplecone}
  Let $k,n$ and $d$ be positive integers such that $2 \leq k < k+2
  \leq n$. Let $m$ be a non-negative integer such that $m+d \geq 3$.
  There is  an injective linear map $$v : Pic_{\QQ}
  (\kgnb{0,m+d}/\mathfrak{S}_d) \rightarrow
  Pic_{\QQ}(\kgnb{0,m}(G(k,n),d))$$ that maps NEF (resp. base-point-free)
  divisors to NEF (resp. base-point-free) divisors. The NEF (resp.
  base-point-free) cone of $\kgnb{0,m}(G(k,n),d)$ is the product of the
  cones spanned by $H_{\sigma_2}, H_{\sigma_{1,1}}, T, L_1, \dots,
  L_m$ and the image under $v$ of the NEF (resp. base-point-free) cone
  of $\kgnb{0,m+d}/\mathfrak{S}_d$.
\end{theorem}

Here the action of the symmetric group $\mathfrak{S}_d$ permutes the
last $d$ marked points. The divisors $H_{\sigma_{1,1}}$ and
$H_{\sigma_2}$ are the divisors of curves whose image intersects fixed
Schubert varieties $\Sigma_{1,1}$ and $\Sigma_2$, respectively.  The
divisor $T$ is the tangency divisor and $L_i$ are the pull-backs of
the ample generator of $Pic(G(k,n))$ by the $i$-th evaluation
morphism. \smallskip


There is a natural way of generating NEF divisors on the Kontsevich
moduli spaces to arbitrary targets $X$ from NEF codimension one and
two classes on $X$. For a large class of target varieties $X$, these
natural NEF divisors together with the pull-back of NEF divisors from
the Deligne-Mumford moduli space generate the NEF cone of
$\Kgnb{0,m}(X, \beta)$. Our proof of Theorem \ref{amplecone} will
remain valid in a more general setting---proving, in particular, the
corresponding statement for flag varieties. \smallskip

In contrast, the effective cones of Kontsevich moduli spaces are
harder to understand and closely depend on the target variety. We
first prove that the effective cones of $\Kgnb{0,0}(G(k,n),d)$
stabilize as long as $n \geq k+d$. We then give a complete description
of the effective cone of $\Kgnb{0,0}(G(k,k+d),d)$. Our methods bound
the effective cone of $\Kgnb{0,0}(G(k,n),d)$ when $n < k+d$; however,
we do not have a complete description of the cone in this case. \medskip

In order to describe the effective cone we have to construct several
effective divisors.  Surprisingly the construction depends on whether
$k$ divides $d$. When $k$ divides $d$, let $\Dunb$ be the class of the
divisor of maps for which the pull-back of the tautological bundle of
$G(k, k+d)$ has non-balanced splitting. If $k$ does not divide $d$,
set $d= kq+r$ for $0 < r < k$. A rational curve in a Grassmannian
induces a rational scroll in projective space.  $\Dunb$ is the class
of the closure of maps where the linear spans of the directrices of
corresponding scrolls intersect a fixed codimension $(k-r)(q+1)$
linear space. $\Ddeg$ is the class of the divisor of maps whose image
lies in a subgrassmannian $G(k,k+d-1)$. With this notation, our second
main theorem is:

\begin{theorem}\label{effth}
  The effective cone of $\Kgnb{0,0}(G(k,k+d),d)$ is a simplicial cone
  generated by $\Ddeg$, $\Dunb$ and the boundary divisors.
\end{theorem}


The organization of the paper is as follows. In \S \ref{secprelim} we
collect the facts that we use about Kontsevich moduli spaces and state
our results more precisely. In \S \ref{secstab} we prove that the
effective cones of $\Kgnb{0,0}(G(k,n),d)$ stabilize when $n \geq k+d$.
In \S \ref{secdiv} we express the divisor classes $\Ddeg$ and $\Dunb$
in terms of the standard basis of the Picard group of
$\Kgnb{0,0}(G(k,k+d),d)$. In \S \ref{secample} we prove a
generalization of Theorem \ref{amplecone}. In \S \ref{pfmain} we prove
Theorem \ref{effth}.  \medskip

\noindent {\bf Acknowledgments:} This work grew out of conversations
with Joe Harris. We are grateful to him for his suggestions and
encouragement.  

\section{Preliminaries}\label{secprelim}

In this section we recall the necessary facts about Kontsevich moduli
spaces. \medskip

Let $X$ be a smooth, complex projective homogeneous variety. Let
$\beta \in H_2 (X, \ZZ)$ be the homology class of a curve. The
Kontsevich moduli space of $m$-pointed, genus-zero stable maps
$\Kgnb{0,m} (X, \beta)$ provides a useful compactification of the
space of rational curves on $X$ whose homology class is $\beta$.
$\Kgnb{0,m} (X, \beta)$ is the smooth, proper Deligne-Mumford stack
parameterizing the data of
\begin{enumerate}
\item[(i)]
  $C$, a proper, connected, at-worst-nodal curve of arithmetic genus
  $0$,
\item[(ii)]  
  $p_1,\dots,p_m$, an ordered sequence of distinct, smooth points of
  $C$,
\item[(iii)] and $f:C\rightarrow X$, a morphism with $f_*[C] = \beta$
  satisfying the following stability condition: every irreducible
  component of $C$ mapped to a point under $f$ contains at least $3$
  special points, i.e., marked points $p_i$ and nodes of $C$.
\end{enumerate}
By \cite{pandh:connected} $\Kgnb{0,m} (X, \beta)$ is irreducible.
\medskip



Oprea in \cite{dragos:divisors} proves that the rational complex
codimension one classes on $\Kgnb{0,m} (X, \beta)$ are tautological.
When $X$ is the Grassmannian $G(k,n)$, one can very explicitly
describe the Picard group $Pic_{\QQ}(\Kgnb{0,0} (G(k,n), d)$ as
follows.  The cohomology ring of the Grassmannian $G(k,n)$ is
generated by the Poincar\'e duals of the classes of Schubert
varieties. We use the convention in \cite{joe:thebook} to denote
Schubert varieties. There are two divisor classes $H_{\sigma_{1,1}}$
and $H_{\sigma_{2}}$ on $\Kgnb{0,0} (G(k,n), d)$ corresponding to the
two codimension-two classes $\sigma_{1,1}$ and $\sigma_2$ on $G(k,n)$.
Informally, these are the classes of the divisors that parameterize
the maps whose images intersect fixed Schubert varieties
$\Sigma_{1,1}$ and $\Sigma_{2}$, respectively. More precisely, let
$$ev_i :\Kgnb{0,m} (G(k,n), d) \rightarrow G(k,n)$$ denote the $i$-th
evaluation morphism and let
$$\pi: \Kgnb{0,1} (G(k,n), d) \rightarrow \Kgnb{0,0} (G(k,n), d)$$
denote the forgetful morphism. $H_{\sigma_{1,1}}$ and $H_{\sigma_{2}}$
are defined as follows:

$$H_{\sigma_{1,1}} := \pi_* \ ev_{1}^* (\sigma_{1,1}), \ \ \
H_{\sigma_{2}} := \pi_* \ ev_{1}^* (\sigma_{2}).$$
In addition to these, for every $1 \leq i \leq \lfloor d/2 \rfloor$,
there is a boundary divisor on $\Kgnb{0,0} (G(k,n), d)$ whose general
point parameterizes a map with a reducible domain $C_1 \cup C_2$ where
the map has Pl\"ucker degree $i$ on $C_1$ and degree $d - i $ on
$C_2$, respectively. We will denote the class of the boundary divisor
by $\Delta_{i,d-i}$.  The theorem of Oprea reduces to the following in
the special case of $G(k,n)$.

\begin{theorem}\label{Pic}(\cite{dragos:divisors}) Let $2 \leq k < k+2
  \leq n$ and $d \geq 1$ be integers.  The divisor classes
  $H_{\sigma_{1,1}}, H_{\sigma_2}$ and $\Delta_{i,d-i}$ for $1 \leq i
  \leq \lfloor d/2 \rfloor$  generate $\mbox{Pic}_{\QQ}(\Kgnb{0,0}
  (G(k,n), d)) $.
\end{theorem}
\medskip


\noindent Let $P_{k,n,d}$ be the $\QQ$-vector space spanned by basis
elements labelled by $H_{\sigma_{1,1}}, H_{\sigma_2}$ and
$\Delta_{i,d-i}$ for $1 \leq i \leq \lfloor d/2 \rfloor$. $\mbox{Pic}_{\QQ}
(\Kgnb{0,0} (G(k,n), d))$ can be naturally identified with
$P_{k,n,d}$. Let $\mbox{Eff}\ (k,n,d)$ denote the image of the
effective cone under this identification.  We can view the effective
cones $\mbox{Eff}\ (k,n,d)$ for different $n$ as cones in a fixed
vector space since $P_{k,n,d}$ does not depend on $n$.  The first
basic result about the cones $\mbox{Eff}\ (k,n,d)$ is that if we fix
$k$ and $d$, they stabilize for large $n$.


\begin{theorem}\label{stability}
For integers $k \geq 2$ and $r \geq 2$, the cone $\mbox{Eff}\ (k,k+r,d)$
is contained in the cone $\mbox{Eff} \ (k,k+r+1,d)$. Furthermore,
$\mbox{Eff}\ (k,k+r,d)= \mbox{Eff} \ (k,k+d,d)$ for every $r \geq d$.
\end{theorem}
\noindent In view of Theorem \ref{stability} it is especially
interesting to describe $\mbox{Eff}\ (k,k+d,d)$. For the rest of the
paper we will concentrate on this case.  On $\Kgnb{0,0} (G(k,k+d),d)$
there are two natural effective $\QQ$-Cartier divisors
$D_{\mbox{deg}}$ and $D_{\mbox{unb}}$. Their classes will play an
essential role in the description of the effective cone of $\Kgnb{0,0}
(G(k,k+d),d)$. \smallskip


Let $D_{\mbox{deg}}$ be the class of the divisor of maps whose image
lies in a subgrassmannian $G(k,k+d-1)$ induced from a
$k+d-1$-dimensional linear subspace. This divisor is the analogue of
the divisor of degenerate maps $D_{deg}$ on $\Kgnb{0,0} (\PP^d,d)$
(see \S 2.2 of \cite{coskun:effective}). We will calculate the class
$D_{\mbox{deg}}$ in \S \ref{secdeg}. \medskip



The definition of $D_{\mbox{unb}}$ depends on whether $k$ divides $d$.  If
$k$ divides $d$, then a general point in $\Kgnb{0,0} (G(k,k+d),d)$
parameterizes a map $f: \PP^1 \rightarrow G(k,k+d)$ such that the
pull-back of the tautological bundle from $G(k,k+d)$ has balanced
splitting. In codimension one the pull-back of the tautological bundle
does not have balanced splitting. Informally the class of the closure
of the subvariety where the splitting is not balanced is
$D_{\mbox{unb}}$. More precisely, let $F(k-1,k,k+d)$ denote the
two-step flag variety parameterizing $k-1$-dimensional linear
subspaces contained in $k$-dimensional linear subspaces of a fixed
$k+d$-dimensional linear space. The flag variety $F(k-1,k,k+d)$ admits
two projections $\pi_1$ and $\pi_2$ to the Grassmannians $G(k-1, k+d)$
and $G(k,k+d)$, respectively. A curve class in $F(k-1,k,k+d)$ is
determined by specifying two integers $d_1, d_2$, where $d_i$ is the
intersection of the curve class with the pull-back of the ample
generator by $\pi_i$. \medskip

Consider the Kontsevich moduli space $\Kgnb{0,0}(F(k-1,k,k+d),
(d-k-1,d))$. The second projection $\pi_2$ induces a 1-morphism
$$\Kgnb{0,0} (\pi_2) : \Kgnb{0,0}(F(k-1,k,k+d), (d-k-1,d))\rightarrow
\Kgnb{0,0} (G(k,k+d),d).$$ The morphism $\Kgnb{0,0} (\pi_2)$ is
birational onto its image. Since $\Kgnb{0,0}(F(k-1,k,k+d), (d-k-1,d))$
is irreducible, the image is irreducible.  Moreover, an easy dimension
count shows that the image has codimension one in $\Kgnb{0,0}
(G(k,k+d),d)$. It follows that the image is an irreducible divisor. We
denote its class by $D_{\mbox{unb}}$.
\medskip
 

If $k$ does not divide $d$, then the locus of maps where the splitting
type of the tautological bundle is not balanced has codimension two,
hence does not form a divisor. In that case, we can write $d = kq + r$
where $0 < r < k$. Let $F(k-r,k,k+d)$ be the two-step flag variety of
$(k-r)$-dimensional subspaces contained in $k$-dimensional subspaces of a
fixed $(k+d)$-dimensional linear space. The second projection $$\pi_2:
F(k-r,k,k+d) \rightarrow G(k,k+d)$$ induces a birational morphism
$$\Kgnb{0,0}(\pi_2) : \Kgnb{0,0}(F(k-r,k,k+d), ((k-r)q,d))\rightarrow
\Kgnb{0,0} (G(k,k+d),d).$$ The first projection $$\pi_1: F(k-r,k,k+d)
\rightarrow G(k-r,k+d)$$ induces a 1-morphism $$\Kgnb{0,0}(\pi_1) :
\Kgnb{0,0}(F(k-r,k,k+d), ((k-r)q,d))\rightarrow \Kgnb{0,0}
(G(k-r,k+d),(k-r)q).$$ Let $S$ denote the tautological bundle of
$G(k-r,k+d)$. $\PP S$ has natural projections $\phi_1$ and $\phi_2$ to
$G(k-r,k+d)$ and $\PP^{k+d-1}$, respectively. By the universal
property of $\PP S$, the evaluation morphism $$ev: \Kgnb{0,1}
(G(k-r,k+d),(k-r)q) \rightarrow G(k-r,k+d)$$ induces a 1-morphism to
$\PP S$ and hence by composing with $\phi_2$ to $\PP^{k+d-1}$. Asking
for the linear span of the image to intersect a fixed $k+d-1 -
(k-r)(q+1)$-dimensional projective linear subspace of $\PP^{k+d-1}$
defines a class in $\Kgnb{0,1} (G(k-r,k+d),(k-r)q)$ whose push-forward
by the forgetful morphism is a divisor $D$ on $\Kgnb{0,0}
(G(k-r,k+d),(k-r)q)$.  We define $\Dunb$ as $$\Dunb =
\Kgnb{0,0}(\pi_2)_* \ \Kgnb{0,0}(\pi_1)^* D.$$ With this notation, we
can rephrase Theorem \ref{effth} as follows:




\begin{theorem}\label{main}
A divisor $D$ on $\Kgnb{0,0} (G(k,k+d),d)$ lies on the effective cone
if and only if $D$ is a non-negative linear combination $$D = a \
\Ddeg + b \ \Dunb + \sum_{i=1}^{\lfloor d/2 \rfloor}
c_i \Delta_{i,d-i}, \ \ \ \ a, b, c_i \geq 0$$ of the divisors
$\Ddeg$, $\Dunb$ and the boundary divisors.
\end{theorem}

The proof is similar to the proof of the main theorem in
\cite{coskun:effective}. Recall that a moving curve is a reduced,
irreducible curve whose deformations cover a Zariski open subset of
$\Kgnb{0,0} (G(k,k+d),d)$. Every moving curve in $\Kgnb{0,0}
(G(k,k+d),d)$ gives rise to an inequality on the effective cone.  If
$B$ is a moving curve and $D$ is an effective Cartier divisor, then $B
\cdot D \geq 0$. Since the divisors $D_{\mbox{deg}}$, $D_{\mbox{unb}}$
and the boundary divisors are effective, any non-negative linear
combination is also effective. The proof is completed by constructing
enough moving curves such that the inequalities they give rise to cut
out precisely the cone spanned by $D_{\mbox{deg}}$, $D_{\mbox{unb}}$
and the boundary divisors. \medskip


The following lemma gives an easy criterion for deciding whether a
reduced, irreducible curve $B \subset \Kgnb{0,0} (G(k,k+d),d)$ is a
moving curve.

\begin{lemma}\label{moving}
A reduced, irreducible curve $B \subset \Kgnb{0,0} (G(k,k+d),d)$ is a
moving curve if $B$ contains a point parameterizing a map $$f: C
\rightarrow G(k,k+d),$$ where the domain of $C$ is irreducible, the
image of $f$ is non-degenerate and the pull-back of the tautological
bundle of $G(k,k+d)$ to $C$ has balanced splitting.
\end{lemma}

\begin{proof}
Let $f: C \rightarrow G(k,k+d)$ be a map satisfying the assumptions of
the lemma. The orbit of this map under the action of $\PP GL (k+d)$
forms a dense subset of $\Kgnb{0,0} (G(k,k+d),d)$. Consequently, the
$\PP GL(k+d)$ orbit of $B$ covers a Zariski open subset of $\Kgnb{0,0}
(G(k,k+d),d)$. Therefore, $B$ is a moving curve.
\end{proof}
\medskip




\section{The stability of the effective cone}\label{secstab}

In this section we prove that the effective cones of
$\Kgnb{0,0}(G(k,n),d)$ stabilize when $n \geq k+d$. This proves
Theorem \ref{stability} stated in \S \ref{secprelim}. 
\medskip



\begin{proof}[Proof of Theorem \ref{stability}]
For the proof it is more convenient to think of $G(k,k+r) =
\gr(k-1,k+r-1)$ as the parameter space for $k-1$-dimensional
projective linear spaces in $\PP^{k+r-1}$. Let $p \in \PP^{k+r}$ be a
point. Let $U =\PP^{k+r} - \{ p \}$ be the complement of the
point. Let $$ \tilde{\pi} : U \rightarrow \PP^{k+r-1}$$ denote linear
projection from the point $p$. The morphism $\tilde{\pi}$ induces a
corresponding morphism $$\pi: V \rightarrow \gr(k-1,k+r-1),$$ where
$V$ denotes the open subscheme of $\gr(k-1,k+r)$ parameterizing
$\PP^{k-1}$ not containing the point $p$. This in turn induces a
smooth 1-morphism $$\Kgnb{0,0}(\pi, k+r) : \Kgnb{0,0}(V,d) \rightarrow
\Kgnb{0,0}(\gr(k-1,k+r-1),d).$$ Let $i : V \rightarrow \gr(k-1,k+r)$
be the open immersion of $V$ into $\gr(k-1,k+r)$. The morphism $i$
also induces a 1-morphism between Kontsevich moduli spaces
$$\Kgnb{0,0}(i,d) : \Kgnb{0,0}(V,d) \rightarrow
\Kgnb{0,0}(\gr(k-1,k+r),d).$$ The complement of the image has
codimension $r$. Since $r \geq 2$, the pull-back morphism on the
Picard groups $$\Kgnb{0,0}(i,d)^* :
\mbox{Pic}_{\QQ}(\Kgnb{0,0}(\gr(k-1,k+r),d))  \rightarrow
\mbox{Pic}_{\QQ}(\Kgnb{0,0}(V,d))$$ is an isomorphism. There is a
unique homomorphism $$h : \mbox{Pic}_{\QQ}(\Kgnb{0,0}(\gr(k-1,k+r
-1),d)) \rightarrow
\mbox{Pic}_{\QQ}(\Kgnb{0,0}(\gr(k-1,k+r),d))$$ such that
$$\Kgnb{0,0}(\pi, k+r)^* = \Kgnb{0,0}(i,d)^* \circ h.$$ Since
$\Kgnb{0,0}(\pi, k+r)$ is smooth it pulls-back effective divisors to
effective divisors. It follows that $\mbox{Eff} \ (k,k+r,d) \subset
\mbox{Eff} \ (k,k+r+1,d)$. \medskip

Next suppose that $r \geq d$. Let $D$ be any effective divisor on
$\Kgnb{0,0}(G(k,k+r,d)$. A general point in the complement of $D$
parameterizes a stable map $f : C \rightarrow G(k,k+r)$ such that the
image lies in a subgrassmannian $G(k,k+d)$. The linear embedding of
the corresponding $\PP^{k+d-1}$ into $\PP^{k+r-1}$ induces a
1-morphism $$\Kgnb{0,0}(k+d,k+r) : \Kgnb{0,0}(G(k,k+d,d) \rightarrow
\Kgnb{0,0}(G(k,k+r,d).$$ By construction $\Kgnb{0,0}(k+d,k+r)^* ([D])$
is the class of the effective divisor
$\Kgnb{0,0}(k+d,k+r)^{-1}(D)$. Therefore, $\mbox{Eff} \ (k,k+d,d)$
contains $\mbox{Eff} \ (k,k+r,d)$. Hence, by the previous paragraph
the two are equal. This concludes the proof of Theorem
\ref{stability}.


\end{proof}


\section{The divisor class calculations}\label{secdiv}


\subsection{The divisor class $D_{\mbox{deg}}$} \label{secdeg} In this
section we express the divisor class $D_{\mbox{deg}}$ in terms of the
standard generators of $\mbox{Pic}_{\QQ}(\Kgnb{0,0} (G(k,k+d),
d))$. We recall that $D_{\mbox{deg}}$ is the class of the
divisor that parameterizes stable maps whose image lies in a proper
subgrassmannian $G(k,k+d-1)$.

\begin{lemma}\label{deg}
The class of the divisor $D_{\mbox{deg}}$ in $\Kgnb{0,0} (G(k,k+d),d)$
can be expressed as $$D_{\mbox{deg}} = \frac{1}{2d} \left( (1-d) \
H_{\sigma_{1,1}} + (d+1)\ H_{\sigma_2} -\sum_{i=1}^{\lfloor d/2
\rfloor} i(d-i) \ \Delta_{i,d-i} \right). $$
\end{lemma}

\begin{proof}
  We determine the class of the divisor $D_{\mbox{deg}}$ by
  intersecting it with test curves. Fix a general rational normal
  scroll of degree $i$ and a rational normal curve $C$ of degree
  $d-i-1$ intersecting the scroll in one point $p$. Consider the
  one-parameter family of degree $d$ rational curves consisting of $C$
  union a pencil (with $p$ as a base point) of rational curves of
  degree $i+1$ on the scroll joined to $C$ along $p$. Take a general
  linear $\PP^{k-2}$ and form the cone over this one-parameter family
  of curves. Each cone gives a one-parameter family of $\PP^{k-1}$,
  hence by the universal property of the Grassmannians induces a curve
  of degree $d$ in $G(k,k+d)$. Denote the resulting one-parameter
  family of degree $d$ curves in $G(k,k+d)$ by $C_i$. The following
  intersection numbers are easy to see: $$C_i \cdot H_{\sigma_{1,1}} =
  0, \ \ \ C_i \cdot H_{\sigma_2} = i, \ \ \ C_i \cdot D_{\mbox{deg}}
  = 0.$$ The curve $C_i$ is contained in the boundary $\Delta_{i+1,
    d-i-1}$. The intersection numbers with the boundary divisors are
  as follows: $$C_i \cdot \Delta_{i+1, d-i-1} = -1, \ \ C_i \cdot
  \Delta_{i,d-i} = 1, \ \ C_i \cdot \Delta_{1,d-1} = i+1,$$ provided
  that $i \geq 1$. The intersection of $C_i$ with the other boundary
  divisors is clearly zero.  \medskip

Next consider the one-parameter family of degree $d$ rational curves
containing $d+2$ general points and intersecting a general line in a
fixed $\PP^d$. Taking a cone over these curves with a fixed general
$\PP^{k-2}$ induces a curve $C$ in $\Kgnb{0,0}(G(k,k+d),d)$ with the
following intersection numbers (see Lemma 2.1 in
\cite{coskun:effective}): $$ C\cdot H_{\sigma_{1,1}} = 0, \ \ \ C \cdot
H_{\sigma_2} = \frac{d^2 + d - 2}{2}, \ \ \ C \cdot \Delta_{1,d-1} =
\frac{(d+2)(d+1)}{2}.$$ The intersection of $C$ with the remaining
boundary divisors is zero. \medskip


Next consider a fixed degree $d-1$ rational normal curve $R$ and a
general pencil of lines based at a point of $R$.  Take a general
$\PP^{k-2}$ and take the cone of the curves with vertex this
$\PP^{k-2}$. This one-parameter family of cones induces a
one-parameter family of rational curves in $G(k,k+d)$. Denote by $B$
the resulting family in $\Kgnb{0,0} (G(k,k+d),d)$. We have the
intersection numbers $$B \cdot H_{\sigma_{1,1}}= 0 , \ \ \ B \cdot
H_{\sigma_2} = 1, \ \ \ B \cdot D_{\mbox{deg}} = 1.$$ $B$ is disjoint
from all the boundary divisors $\Delta_{i,d-i}$ for $i>1$. It is
contained in $\Delta_{1,d-1}$ with intersection number $B \cdot
\Delta_{1,d-1} = -1$. These test curves suffice to determine the class
of $D_{\mbox{deg}}$ except for the coefficient of $H_{\sigma_{1,1}}$.
\medskip

In order to determine the coefficient of $H_{\sigma_{1,1}}$, we take
the following one-parameter family. Fix a rational normal scroll $S$
of dimension $k-1$ and degree $d$. Fix a general line $l$. Consider
the one-parameter family of cones obtained by taking the cone over $S$
with vertex a point of $l$. This family of scrolls induces a curve $A$
in $\Kgnb{0,0} (G(k,k+d),d)$. $A$ has the following intersection
numbers: $$A \cdot H_{\sigma_{1,1}} = d, \ \ A \cdot H_{\sigma_2} = d,
\ \ A \cdot D_{\mbox{deg}} = 1.$$ $A$ is disjoint from all the
boundary divisors. The class of $D_{\mbox{deg}}$ follows form these
intersection numbers.
\end{proof}


\subsection{The divisor class $\Dunb$ when $k$ does not divide $d$} In
this subsection we express the class of the divisor $\Dunb$ in
$\kgnb{0,0}(G(k,k+d),d)$ when $k$ does not divide $d$ in terms of the
standard generators. Express $ d = qk + r$ where $q$ and $r$ are the
unique integers satisfying $0< r <k$. The projective $k-1$-planes in
the image of a general point of $\kgnb{0,0}(G(k,k+d),d)$ sweep out a
$k$-dimensional scroll of degree $d$ in $\PP^{k+d-1}$. Such a scroll
has a unique $k-r$-dimensional minimal subscroll of degree $(k-r)q$.
Recall from \S \ref{secprelim} that the divisor $\Dunb$ is the closure
of the locus where the span of this minimal subscroll intersects a
fixed codimension $(k-r)(q+1)$ linear space. We have already described
the closure explicitly as the image of a divisor from the space of
stable maps to the two-step flag variety under the natural map $$
\Kgnb{0,0}(\pi_2) : \Kgnb{0,0}(F(k-r,k,k+d), ((k-r)q,d))\rightarrow
\Kgnb{0,0} (G(k,k+d),d).$$ Note that $\Dunb$ may be defined in
$\kgnb{0,0}(G(k,n),d)$ provided $n > (k-r)(q+1)$. The expression of
the class in terms of the standard generators is clearly independent of $n$.
\medskip

\begin{lemma}\label{notdiv}
  Let $ d = qk + r$ with $0< r <k$. Assume $n > (k-r)(q+1)$. Let $i =
  q_i k + r_i$ where $q_i$ and $r_i$ are the unique integers
  satisfying $ 0 \leq r_i <k$. Then the divisor class $\Dunb$ in
  $\kgnb{0,0}(G(k,n),d)$ is given by $$\Dunb = -
  \frac{(q+1)(d-q-1)}{2d}H_{\sigma_2} + \frac{(q+1)(d+q+1)}{2d} +
  \sum_{i=1}^{\lfloor d/2 \rfloor} \gamma_i \Delta_{i}$$ where
  $$\gamma_i = \begin{cases} {q_i+1 \choose 2} (r-k) + \frac{q+1}{2d}
    \left(i^2 (q+1) - i d  q_i - d r_i (q_i+1) \right) \ \mbox{if} \
    r_i \leq r \\{q_i+1 \choose 2} (r-k) - (q_i+1)(r_i -r)+
    \frac{q+1}{2d} \left(i^2 (q+1) - i d q_i - d r_i (q_i+1)\right)
 \end{cases}$$ if $r< r_i$.
\end{lemma}

\begin{proof} We will prove the lemma by intersecting both sides of
  the equation by one-parameter families. Since the class of the
  divisor is independent of $n$ we may construct the families in
  $G(k,n)$ for any $n >(k-r)(q+1)$. Fix a balanced rational normal
  scroll of degree $d+q$ and dimension $k+1$. Let $C_q'$ be the
  one-parameter family of subscrolls of degree $d$ and dimension $k$
  obtained by taking the joins of a fixed degree $d-q$ balanced
  rational subscroll of dimension $k-1$ with rational normal curves of
  degree $q$ varying in a 2-dimensional subscroll of degree $2q$.
  Denote by $C_q$ the one-parameter family in $\kgnb{0,0}(G(k,n),d)$
  induced by the family of scrolls. The following intersection numbers
  are easy to compute
  $$C_q \cdot \Dunb = q+1, \ \ C_q \cdot H_{\sigma_2} = d+q, \ \ C
  \cdot H_{\sigma_{1,1}} = d-q.$$ $C_q$ clearly has intersection
  number zero with all the boundary components. \medskip

  Similarly, fix a rational normal scroll of degree $d+q+1$ and
  dimension $k+1$. Let $C_{q+1}'$ be the one-parameter family of
  subscrolls of degree $d$ and dimension $k$ obtained by taking the
  joins of a fixed degree $d-q-1$ balanced rational subscroll of
  dimension $k-1$ with rational normal curves of degree $q+1$ varying
  in a 2-dimensional subscroll of degree $2q+2$. Denote by $C_{q+1}$
  the one-parameter family in $\kgnb{0,0}(G(k,n),d)$ induced by the
  family of scrolls. The following intersection numbers are easy to
  compute: $$C_{q+1} \cdot \Dunb = 0, \ \ C_q \cdot H_{\sigma_2} =
  d+q+1, \ \ C \cdot H_{\sigma_{1,1}} = d-q-1.$$ The intersection of
  $C_{q+1}$ with the boundary divisors is zero. The coefficients of
  $H_{\sigma_2}$ and $H_{\sigma_{1,1}}$ follow from these two
  families. \medskip

  Let $X$ be the blow-up of $\PP^1 \times \PP^1$ at one point. Denote
  by $F_1$ and $F_2$ the pull-backs of the two fiber classes from
  $\PP^1 \times \PP^1$. Let $E$ be the exceptional divisor. Let $V$ 
  be the following vector bundle of rank $k$ on $X$ given by 
$$V = \begin{cases} \oplus_{j=1}^{k-r} \OO_X(-qF_1 - mF_2 + q_i E)
  \oplus \oplus_{j=1}^{r-r_i}\OO_X(-(q+1)F_1 - mF_2 + q_i E)  \\ \oplus
  \oplus_{j=1}^{r_i}\OO_X(-(q+1)F_1 - mF_2 + (q_i+1) E) \ \ \mbox{if}
  \ \ r \geq r_i \\ \oplus_{j=1}^{k-r_i + r} \OO_X(-qF_1 - mF_2 + q_i E)
  \oplus \oplus_{j=1}^{r_i-r}\OO_X(-qF_1 - mF_2 + (q_i+1) E)  \\ \oplus
  \oplus_{j=1}^{r}\OO_X(-(q+1)F_1 - mF_2 + (q_i+1) E) \ \ \mbox{if}
  \ \ r < r_i
\end{cases}$$ Let $\pi:X \rightarrow \PP^1$ denote the natural map to
$\PP^1 \times \PP^1$ followed by the first projection.  Mapping
$\PP(V)$ to projective space under $\OO_{\PP(V)}(1)$ gives a
one-parameter family of scrolls of degree $d$ parameterized by $\PP^1$
which in turn induces a one-parameter family $B_i$ in
$\kgnb{0,0}(G(k,n),d)$. By construction the pull-back of the
tautological bundle of $G(k,n)$ to $X$ is $V$. Hence we have the
following intersection numbers $$B_i \cdot H_{\sigma_2} = c_1^2(V) -
c_2 (V),  \ \ B_i \cdot H_{\sigma_{1,1}} = c_2(V).$$  The Chern
classes of $V$ are determined by the Whitney sum formula to be
$$c_1(V) = -d F_1 - km F_2 + i E$$
$$c_1(V)^2 = 2kmd- i^2$$
\begin{eqnarray*}
c_2(V) &= qm (k-r)(k-r-1) + (q+1)m r(r-1) + (2q+1)mr(k-r) \\ &- q_i^2
{k-r_i \choose 2} - q_i(q_i+1)r_i(k-r_i) - (q_i+1)^2 {r_i \choose 2}\\
&= md(k-1) - \frac{1}{2}(q_i i (k-1) + r_i(i - q_i -1)).
\end{eqnarray*}
Hence, we conclude that $$B_i \cdot H_{\sigma_2}= md(k+1) - i^2 +
\frac{1}{2}(q_i i (k-1) + r_i(i - q_i -1))$$ and 
$$B_i \cdot H_{\sigma_{1,1}} =   md(k-1) - \frac{1}{2}(q_i i (k-1) +
r_i(i - q_i -1)).$$ The intersection number $B_i \cdot \Delta_j =
\delta_{i}^j$, where $\delta_{i}^j$ is the Kr\"onecker delta function.
Finally let $W$ be the subbundles of $V$ giving rise to the minimal
subscrolls of degree $(k-r)q$, i.e., the direct sum of the line bundles
where the coefficient of $F_1$ is $-q$.  The intersection number $B_i
\cdot \Dunb$ is given by $c_1(\pi_*(W^*))$ The latter may be computed
by Riemann-Roch $$B_i \cdot \Dunb= \begin{cases} ((q+1)m - {q_i +1 \choose
    2})(k-r) \ \mbox{if}\ r \geq r_i \\ ((q+1)m - {q_i + 1 \choose 2})(k-r) -
  (q_i+1)(r_i-r) \ \mbox{if}\ r < r_i \end{cases}$$ The lemma follows
by elementary algebra.
\end{proof}



\subsection{The divisor class of $\Dunb$ when $k$ divides $d$}
\label{sec-Dunb}


In this section we express the class of $\Dunb$ in terms of the
standard generators of the Picard group of $\Kgnb{0,0}(G(k,k+d),d)$
when $k$ divides $d$.

\begin{lemma} \label{lem-3.2}
Suppose $k$ divides $d$.  For each integer $i$, 
let $r_i$ be the unique integer $0\leq r_i <k$ such that $k$ divides
$i-r_i$.  The class of the divisor $\Dunb$ satisfies the equation
$$
\Dunb = \frac{1}{2k} \left( (k+1)H_{\sigma_{1,1}} + (1-k)H_{\sigma_2} -
\sum_{i=1}^{\lfloor d/2 \rfloor} r_i(k-r_i) \Delta_{i,d-i} \right).
$$
\end{lemma}

As it occurs, it is simpler to prove a more general result.  

\begin{definition} \label{defn-versal}
A
\emph{versal triple} is a triple
$(S,C,E)$ of
\begin{enumerate}
\item[(i)]
a Deligne-Mumford stack $S$ with quasi-projective coarse moduli space,
\item[(ii)]
a family of prestable, genus $0$ curves $\pi:C\rightarrow S$,
\item[(iii)]
and a locally free sheaf $E$ on $C$ of rank $k$ and degree $d$ 
\end{enumerate}
which is \emph{versal} at every geometric point $s$ of $S$: 
the induced map from the formal neighborhood of $s$
to the versal deformation space of $(C_s,E|_{C_s})$ is (formally) smooth.  
\end{definition}

\begin{lemma} \label{lem-ez}
If the triple $(S,C,E)$ is versal then $S$ is smooth. 
Conversely,
assuming $S$ is
smooth at $s$, $(S,C,E)$ is versal at $s$ if the map from
the Zariski tangent space $T_s S$ to the first-order deformation space
of $(C_s,E_s)$ is surjective.
\end{lemma}

\begin{proof}
Since the
natural 
obstruction group to deformations of $(C_s,E|_{C_s})$ is of 
the form $H^2(C_s,-)$, the versal deformation space is smooth.
Thus $S$ is smooth if $(S,C,E)$ is versal.

Conversely, assume $S$ is smooth.
A $k$-algebra homomorphism between power series rings is
formally smooth if and only if it is surjective on Zariski
tangent spaces.  
\end{proof}

For every versal triple $(S,C,E)$, 
$S$ contains a collection of reduced, effective Cartier
divisors $\Dunb$ and $(\Delta_{i,d-i})_{i\in \ZZ}$ with the following
properties.  
\begin{enumerate}
\item[(i)]
A codimension 1 point $s$ is in $\Dunb$ if and only if
$$
C_s \cong \PP^1, \ \ E|_{C_s} \cong \OO((d/k)-1) \oplus
\OO(d/k)^{\oplus(k-2)} \oplus \OO((d/k)+1).
$$
\item[(ii)]
A codimension 1 point $s$ is in $\Delta_{i,d-i}$ if and only if
$$
C_s = C'_s\cup C''_s, \ \ \text{deg}(E|_{C'_s}) = i, \ \
\text{deg}(E|_{C''_s}) = d-i.
$$
\end{enumerate}
Denote by $U$ the
complement of the divisor $\Dunb$ and all the divisors $\Delta_{i,d-i}$.  
For every geometric point $s$ of $U$
$$
C_s \cong \PP^1, \ \ E|_{C_s} \cong \OO(d/k)^{\oplus k}.
$$

\begin{lemma} \label{lem-3.9}
For every versal triple $(S,C,E)$ there is a linear equivalence of
$\QQ$-Cartier divisor classes on $S$,
$$
2k[\Dunb] \sim \pi_*C_2(\text{End}(E)) - \sum_{i\leq \lfloor d/2
  \rfloor} r_i(k-r_i)[\Delta_{i,d-i}].
$$ 
\end{lemma}

First we deduce Lemma ~\ref{lem-3.2} assuming Lemma ~\ref{lem-3.9}.

\begin{proof}[Proof of Lemma \ref{lem-3.2}]
Form the triple with $S=\Kgnb{0,0}(G(k,k+d),d)$, with
$\pi:C \rightarrow \Kgnb{0,0}(G(k,k+d),d)$ the
universal family of curves, and with
$E= f^*S_k^\vee$. Here $S_k$ is the tautological rank $k$ subbundle on
$G(k+d,d)$ and $f:C \rightarrow G(k,k+d)$ is the universal map.

The stack $S$ is smooth, in fact smooth over the stack $\mfM{0,0}$ of
prestable, genus $0$ curves, cf. \cite{B}.
Using the relative smoothness, Lemma ~\ref{lem-ez} says 
$(S,C,E)$ is versal if each
relative Zariski tangent space of $S$ over $\mfM{0,0}$ surjects to
the first-order deformation space of $E_s$.

The Euler sequence on
$G(k,n)$ is,
$$
\begin{CD}
0 @>>> \text{Hom}(S_k,S_k) @>>> \text{Hom}(S_k,\OO_{G(k,n)}^{\oplus
  n}) @>>> T_{G(k,n)} @>>> 0.
\end{CD}
$$
Pulling back by $f$ and pushing forward by $\pi$ gives an exact sequence
$$
\begin{CD}
\pi_*f^*T_{G(k,n)} @>>> R^1\pi_* \text{Hom}(E,E)
@>>> R^1 \pi_* (f^*S_k^\vee)^{\oplus n}.
\end{CD}
$$
The first term is the relative tangent bundle of
$\Kgnb{0,0}(G(k,k+d),d)$ over $\mfM{0,0}$. 
The second term is the
bundle of first-order deformation spaces of $E$. 
Since $f^* S_k^\vee$ is globally generated,
$R^1\pi_* (f^* S_k^\vee) = (0)$
by ~\cite[Lemma 2.3(i)]{HS2}.  Therefore $(S,C,E)$ is versal.

Clearly the divisors $\Dunb$ and $(\Delta_{i,d-i})_{i\in
  \ZZ}$ associated to $(S,C,E)$ are the same as the divisors 
defined earlier.  Finally, by straightforward
computation 
$$
c_2(\text{End}(S_k)) = (k+1)\sigma_{1,1} + (1-k)\sigma_2.
$$
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem-3.9}]
Let 
$$
\begin{CD}
0 @>>> \omega_\pi @>>> \mc{E}_\pi @>>> \OO_C @>>> 0
\end{CD}
$$
be the extension whose class 
in $H^1(C,\omega_\pi)$ defines the trace.
Form the locally free sheaf $\mc{F} = \text{End}(E)\otimes \mc{E}_\pi$.
For every geometric point $s$ of $U$
$$
\mc{F}|_{C_s} = (\OO(-1)\oplus\OO(-1))\otimes
\text{End}(\OO(d/k)^{\oplus k}) \cong \OO(-1)^{\oplus 2k^2}.
$$
Since $\OO(-1)$ has no cohomology
$$
h^0(C_s,\mc{F}|_{C_s}) = h^1(C_s,\mc{F}|_{C_s}) = 0.
$$
Since $U$ is a dense Zariski open, $\pi_*\mc{F} = (0)$ and
$R^1\pi_*\mc{F}$ is a torsion sheaf supported on
$\Dunb$ and the divisors $\Delta_{i,d-i}$.  The determinant of this
torsion sheaf
is an effective Cartier
divisor $R$ supported on $\Dunb$ and the divisors $\Delta_{i,d-i}$,
cf. \cite{detdiv}.
Thus
$$
R = r_{\mbox{unb}} \Dunb + \sum_{i\leq \lfloor d/2 \rfloor} r_{i,d-i}
\Delta_{i,d-i}. 
$$
If $\Dunb$ is nonempty
$r_{\mbox{unb}}$
is the length of the stalk
$(R^1\pi_* \mc{F})_s$ at a generic point $s$ of $\Dunb$.  Similarly
for $r_{i,d-i}$.   

Because the formal neighborhood
of $s$ in $S$ maps smoothly to the versal deformation space of
$(C_s,E_s)$, the length of $(R^1\pi_*\mc{F})_s$ 
can be computed using the versal deformation space
and thus using any other versal triple.  In other words, there exist
integers $r_{\mbox{unb}}$ and $r_{i,d-i}$ such that the equation above
holds for \emph{every} versal triple $(S,C,E)$.  
This allows us to
compute the coefficients $r_{\mbox{unb}}$ and $r_{i,d-i}$ using
special versal triples, i.e., using the method of \emph{test
  families}.

Also Riemann-Roch gives 
$$
R \sim 2\pi_* c_2(\text{End}(E)),
$$
cf. \cite{dJS3}.
This gives a
linear equivalence of $\QQ$-Cartier classes
$$
2\pi_* c_2(\text{End}(E)) \sim r_{\mbox{unb}} \Dunb + \sum_{i\leq \lfloor
  d/2 \rfloor} r_{i,d-i} \Delta_{i,d-i}.
$$

Let $V$ and $W$ be $2$-dimensional vector spaces.
Set $S=\PP V \cong \PP^1$.  Let $C=\PP
V \times \PP W \cong \PP^1 \times \PP^1$ 
and let $\pi:C\rightarrow S$ be the projection
$\text{pr}_V$.  

The relative dualizing sheaf of $\pi$ is
$\text{pr}_W^*\omega_{\PP(W)}$.  And the sheaf $\mc{E}_\pi$ is,
$$
\mc{E}_\pi = \text{pr}_W^*(W^\vee\otimes_{\QQ} \OO_{\PP(W)}(-1)).
$$  

\medskip\noindent
\textbf{1. Computation of $r_{\mbox{unb}}$.}
There is a natural isomorphism of vector spaces,
$$
H^1(\PP V \times \PP W ,\text{pr}_V^*\OO(1)\otimes
\text{pr}_W^* \omega_{\PP W }) \cong $$
$$
H^0(\PP V ,\OO(1))\otimes H^1(\PP W ,\omega_{\PP W }) \cong
V^\vee.
$$
Let $x$ be a nonzero element in $V^\vee$.  Denote by $p$ the unique
point of $\PP V $ where $x$ vanishes.
Using the isomorphism
$$
\text{Ext}^1(\text{pr}_W^*(\OO(1)), \text{pr}_V^*(\OO(1))
\otimes \text{pr}_W^*(\omega_{\PP W }(1))) \cong 
$$
$$
H^1(\PP V \times
\PP W , \text{pr}_V^*\OO(1)\otimes \text{pr}_W^*
\omega_{\PP W }),
$$
the element $x$ determines an extension class
$$
\begin{CD}
0 @>>> \text{pr}_V^*\OO(1)\otimes
\text{pr}_W^*(\omega_{\PP W}(1)) @>>> G_x @>>>
\text{pr}_W^*\OO(1) @>>> 0.
\end{CD}
$$
Because the section $x$ of $\OO(1)$
is nonzero on $D_+(x)$,
the restriction of $G_x$
to $D_+(x)$ is balanced.  However, the restriction of $G_x$ to
the fiber over $p$ is unbalanced, $\OO(-1)\oplus
\OO(1)$.  

\medskip\noindent
Define locally free sheaves $E'$ and $E$ to be 
$$
E' :=
G_x \oplus \OO_C^{\oplus (k-2)}, \ \ E := E'\otimes 
\text{pr}_W^*\OO(d/k).  
$$
The
datum $(S,C,E)$ is versal on $D_+(x)$.  We claim it
is also versal at $p$, i.e., the induced morphism $\zeta$ from the formal
neighborhood of $p$ to the versal deformation space is smooth.
%% More generally, if $d$ is not $0$,
%% replace $\mc{E}$ by $\mc{E}\otimes \text{pr}_W^*\OO_{\PP(W)}(d/k)$ to
%% get a 1-morphism $\zeta:S\rightarrow \mfF{k,d}$.  
The versal deformation space is $k\Sem{t}$, and the Cartier divisor
$\Dunb$ over the deformation space is just $(t)$.  The pullback
$\zeta^{-1}(\Dunb)$ equals $m\{p\}$ for some
positive integer $m$.  By Lemma ~\ref{lem-ez}, $\zeta$ is smooth if
$m = 1$.
Clearly $m$ is the same for both the triple $(S,C,E)$ and
the triple $(S,C,E')$.

Form the sheaf $\mc{T}_S := R^1\pi_*(E'\otimes \mc{E}_\pi)$, and form
the analogous sheaf $\mc{T}_{\text{versal}}$ on $k\Sem{t}$.  
As above, $\pi_*(E'\otimes \mc{E}_\pi) = (0)$ and $\mc{T}_S$ 
is a torsion sheaf supported on $p$.
Since $\zeta$ is flat and since 
$R\pi_*(E'\otimes \mc{E}_\pi)$ is of
formation compatible with flat base-change,
$$
\text{length}(\mc{T}_S) = m\cdot \text{length}(\mc{T}_{\text{versal}}) \geq
m \cdot \text{dim}(\mc{T}_{\text{versal}}/t\mc{T}_{\text{versal}}) =
m\cdot \text{dim}(\mc{T}_S\otimes \kappa(p)).  
$$
By cohomology and base change
$$
\mc{T}_S\otimes \kappa(p) \cong
H^1(\PP W,[\omega_{\PP W}(1)\oplus \OO_{\PP W }^{\oplus (k-2)}\oplus
\OO_{\PP W}(+1)] \otimes [W^\vee(-1)])
$$
which is canonically $W^\vee$.  Since this is $2$-dimensional
$$
\text{length}(\mc{T}_S) \geq 2m.
$$
Since $\mc{T}_S$ is a torsion sheaf, $\text{length}(\mc{T}_S) =
c_1(\mc{T}_S)$.   
By ~\cite[Lemma 4.3]{dJS3}
$$
c_1(\mc{T}_S) = -c_1(R\pi_*(E'\otimes \mc{E}_\pi)) = -\pi_*(c_1(E')^2-2c_2(E')).
$$
By the Whitney sum formula
$$
c_t(E') = c_t(G_x) =
1+\text{pr}_V^* h_V +(\text{pr}_V^*h_V)\cap(\text{pr}_W^*h_W),
$$
where $h_V=c_1(\OO_{\PP(V)}(1))$ and $h_W = c_1(\OO_{\PP(W)}(1))$.
Hence
$$
\text{length}(\mc{T}_S) =
2\pi_*((\text{pr}_V^*h_V)\cap(\text{pr}_W^*h_W)) = 2.
$$
Since $2 = \text{length}(\mc{T}_S) \geq 2m$, 
$m = 1$, i.e., $(S,C,E)$ is versal.

Because $(S,C,E)$ is
versal, because all the divisors $\Delta_{i,d-i}$ are empty, and
because $\Dunb$ is the single reduced point $p$
$$
r_{\mbox{unb}} = \text{deg}_S(R).
$$
By Riemann-Roch, $R \sim 2\pi_* c_2(\text{End}(E))$.
Of course $\text{End}(E) \cong \text{End}(E')$.  
By straightforward computation,
$$
2c_2(\text{End}(E')) =
-2(\text{rk}(E')-1)c_1(E')^2 + 4\text{rk}(E')c_2(E').
$$
Using the computation above, this is,
$$
-2(k-1)\text{pr}_V^*(h_V\cap h_V) +4k(\text{pr}_V^*h_V)\cap
(\text{pr}_W^*h_W) = 4k(\text{pr}_V^*h_V)\cap (\text{pr}_W^* h_W).
$$
The pushforward to $\PP V$ has degree $4k$;
$$
r_{\mbox{unb}} = 4k.
$$

\medskip\noindent
\textbf{2. Computation of $r_{i,d-i}$.}
Fix a point $p$ in $\PP V$ and a point $p'$ in $\PP W$.
Let $\nu:\widetilde{C}\rightarrow C$ be the blowing-up of
$\PP V\times \PP W$ along $\{(p,p')\}$.  Denote by $A$ the
exceptional divisor of $\nu$.
Denote by
$\widetilde{\pi}:\widetilde{C}\rightarrow S$ the composition
$\text{pr}_V\circ \nu$.  Let $i\leq \lfloor d/2 \rfloor$ be an
integer.  Write,
$$
i = kq_i + r_i,
$$
where $r_i$ is the unique integer $0\leq r_i < k$ 
such that $i-r_i$ is divisible by
$k$. 
Denote
$$
E' := \OO_{\widetilde{C}}(-(q_i+1)A)^{\oplus r_i} \oplus
\OO_{\widetilde{C}}(-q_i A)^{\oplus (k-r_i)}, \ \ E := E'\otimes
\nu^*\text{pr}_W^* \OO(d/k).
$$
Consider the triple $(S,\widetilde{C},E)$.

It is clearly versal on
$\PP V-\{p\}$. 
To see that it is versal at $p$, observe first
the versal
deformation space of $(\widetilde{C}_p,E|_{\widetilde{C}_p})$
equals the versal 
deformation space $\widetilde{C}_p$, which is $k\Sem{t}$.
Let $m$ be the positive integer such that $(t)$ pulls back to $m\{p\}$.
As in \cite[p. 81]{DM},
there is an isomorphism of the formal neighborhood of the
node of $\widetilde{C}_p$ in $\widetilde{C}$ with the
formal scheme $k\Sem{u,v,s}/\langle uv- s^m \rangle$, i.e., an
$A_{m-1}$-singularity.  Since $\widetilde{C}$ is smooth, $m=1$, i.e.,
$(S,C,E)$ is versal.  

As above, $r_{i,d-i}$ equals the degree of the
first Chern class of $-R{\widetilde{\pi}}_*(\text{End}(E')\otimes
\mc{E}_{\widetilde{\pi}})$, 
$$
r_{i,d-i} = 2\widetilde{\pi}_*c_2(\text{End}(E')).
$$
Of course
$$
\text{End}(E') = (E')^\vee\otimes E' =
\OO_{\widetilde{C}}^{\oplus r_i^2} \oplus
\OO_{\widetilde{C}}(-A)^{\oplus r_i(k-r_i)} \oplus
\OO_{\widetilde{C}}(+A)^{\oplus r_i(k-r_i)} \oplus
\OO_{\widetilde{C}}^{\oplus (k-r_i)^2}.
$$
By the Whitney sum formula
$$
c_t(\text{End}(E')) 
= (1-[A]\cap[A])^{r(i)(k-r(i))}.
$$
Therefore
$$
r_{i,d-i} = -2\text{deg}_S \widetilde{\pi}_*(r_i(k-r_i)[E]\cap[E]) =
2r_i(k-r_i)\text{deg}_S(\{p\}) = 2r_i(k-r_i). 
$$
\end{proof}


\section{The ample cone} \label{secample}

In this section we prove a generalization of Theorem \ref{amplecone}
for products of flag varieties. Our proof will be valid in more
general contexts; however, for simplicity of exposition we leave
generalizations to the reader.  \medskip

Let $X$ be a product of flag varieties with Picard number $\rho$.
There are $\rho$ morphisms from $X$ to Grassmannians $\phi_i: X
\rightarrow G_i$. The cone of NEF divisors on $X$ is simplicial and
generated by the pull-backs of the ample divisors on $G_i$ by the
morphisms $\phi_i$. We denote these divisors by $H_1, \dots,
H_{\rho}$. Let $\beta \in H_2(X, \ZZ)$ be a curve class. For our
purposes we may and will assume that $\beta \cdot H_i = e_i >0 $ for
every $1 \leq i \leq \rho$ (otherwise, we can reduce to the problem of
understanding the ample cone when the target has smaller Picard
number). \medskip

The NEF divisors on $X$ pull-back to NEF divisors on $\Kgnb{0,m}(X,
\beta)$ via the evaluation morphisms (see \cite{coskun:nef} Lemma
4.1). We thus obtain the NEF divisors $L_i^j = ev_i^*(H_j)$ on
$\Kgnb{0,m}(X, \beta)$ for $1 \leq i \leq m $ and $ 1 \leq j \leq
\rho$. \medskip


The tangency divisor $T$ plays an important role in the description of
the ample cone. $T$ is the class of the locus of maps for which the
inverse image of a fixed hyperplane in the Pl\"ucker embedding of
$G(k,n)$ does not consist of $d$ distinct, smooth points of the
domain. For this class to make sense we need that $d> 1$.  R.
Pandharipande in \cite{pandh:qdivisors} computes the class of the
tangency divisor when the target is projective space. Since $T$ on
$\Kgnb{0,m} (G(k,n), d))$ is the pull-back of this divisor by the
Pl\"ucker embedding, the following formula easily follows from
Pandharipande's calculation in \cite{pandh:qdivisors} Lemma 2.3.1:
$$T = \frac{d-1}{d}H_{\sigma_{1,1}} + \frac{d-1}{d}H_{\sigma_2} +
\sum_{i=1}^{\lfloor d/2 \rfloor} \frac{i(d-i)}{d} K_i, $$ where $K_i$
is the sum of the boundary divisors where the map has degree $i$ on
one component. The divisor $T$ is NEF on $\Kgnb{0,m} (G(k,n), d))$
(\cite{coskun:nef} Lemma 4.1). Assuming that $e_i > 1$, pulling-back
the tangency divisor on $\kgnb{0,m}(G_i,e_i)$ by $\Kgnb{0,m}(\phi_i)$
we obtain NEF divisors $T_1, \dots, T_{\rho}$ on $\Kgnb{0,m}(X,
\beta)$. \medskip

The numerically effective codimension two cycles on $X$ are also
simplicial and give rise to NEF divisors on $\Kgnb{0,m}(X, \beta)$.
Given a cycle $\gamma \in NEF^2(X)$, the divisor $$H_{\gamma} :=
\pi_{{m+1}*} (ev_{m+1}^*(\gamma))$$ is NEF (\cite{coskun:nef} Lemma
4.1), where $ev_{m+1}$ is the $m+1$-st evaluation morphism from
$\Kgnb{0,m+1}(X, \beta)$ to $X$ and $\pi_{m+1}$ is the forgetful
morphism. \medskip

Let $\kgnb{0,m+e_1+ \cdots + e_\rho}/\mathfrak{S}_{e_1} \times \cdots
\times \mathfrak{S}_{ e_\rho}$ denote the quotient of the moduli space of
genus-zero stable curves with $m+e_1 + \cdots + e_{\rho}$ marked
points by the action of the products of the symmetric groups on $e_1,
\dots, e_{\rho}$ letters acting by the obvious permutation action.
There is a map
$$v: \mbox{Pic}_{\QQ}(\kgnb{0,m+e_1+ \cdots + e_\rho}/\mathfrak{S}_{e_1}
\times \cdots \times \mathfrak{S}_{ e_\rho}) \rightarrow
\mbox{Pic}_{\QQ}(\Kgnb{0,m} (X, \beta))$$ that maps NEF divisors to
NEF divisors. This map is constructed and described in \S 2 of
\cite{coskun:nef} for the case of $\PP^n$. The proofs of that section
carry over to the case of $X$. The map $v$ is induced by the rational
map from $\Kgnb{0,m} (X, \beta)$ to $\kgnb{0,m+e_1+ \cdots +
  e_\rho}/\mathfrak{S}_{e_1} \times \cdots \times \mathfrak{S}_{ e_\rho}$
obtained by intersecting the image of the map with a fixed union of
divisors $H_1 \cup \cdots \cup H_{\rho}$, marking the points of
intersection and then forgetting the map and stabilizing.


\begin{theorem}\label{ample}
  The NEF cone of $\Kgnb{0,m} (X, \beta)$ is the product of the image
  under $v$ of the NEF cone of $\kgnb{0,m+e_1+ \cdots +
    e_\rho}/\mathfrak{S}_{e_1} \times \cdots \times \mathfrak{S}_{
    e_\rho}$ with the cone spanned by non-negative linear combinations
  of the classes $L_i^j$, for $1 \leq i \leq m $ and $ 1 \leq j \leq
  \rho$, $H_{\gamma}$, for NEF codimension two cycles $\gamma$ on $X$,
  and $T_i$, for $1 \leq i \leq \rho$ and $e_i > 1$.
\end{theorem}

\begin{proof} The proof is almost identical to the proof of the main
  theorem in \cite{coskun:nef}. Clearly any divisor $D$ that lies in
  this cone is NEF. The main theorem of \cite{dragos:divisors} and
  Proposition 2.5 of \cite{coskun:nef} imply that divisors in this
  cone span the Picard group. Hence, we can write any NEF divisor as
  $$D = v(D_1) + \sum_{\gamma \in NEF^2(X)} a_{\gamma} H_{\gamma} + 
  \sum_{i=1}^{\rho} b_i T_i + \sum_{1\leq i \leq \rho, 1\leq j \leq m}
  c_{i,j} L_i^j.$$ We have to show that $D_1$ is NEF and the constants
  $a_{\gamma}, b_i $ and $c_{i,j}$ are non-negative. We can do this by
  intersecting with curves that have zero intersection with all the
  terms in the expression of $D$ but one. Analogues of the necessary
  curves have appeared in \cite{coskun:nef}.  \medskip

  Briefly take any curve $C$ in $\kgnb{0,m+e_1 + \cdots
    +e_{\rho}}/\mathfrak{S}_{e_1} \times \cdots\mathfrak{S}_{e_{\rho}}
  $. Attach $e_i$ copies of the generator $B_i$ of the curve class in
  $G_i$ to the $e_i$ marked points of the members of $C$. Map the
  curves $B_i$ isomorphically to fixed representatives passing through
  a point and contract the members of $C$.  This curve has zero
  intersection with $T_i, H_{\gamma}$ and $L_i^j$. Hence $D_1$ has to
  be NEF. \medskip

To see that the coefficients of $L_i^j$ are non-negative take a
 curve in the class $\beta$ with two components $C_1 \cup C_2$ where
 the component $C_1$ has class dual to $H_j$
 and $C_2$ has the residual class. Suppose $C_2$ contains all the
 marked points but the $i$-th marked point. Consider the one-parameter
 family obtained by varying the $i$-th marked point $p_i$. This
 one-parameter family has positive intersection with  $L_i^j$ and zero
 intersection with all the other divisors occurring in the expression
 of $D$. Hence $c_{i,j} \geq 0$. \medskip
     

 To prove that the coefficients of $T_i$ are non-negative consider the
 following one-parameter family. Take a fixed curve $C$ with $m$
 marked points in the class $\beta - 2\zeta_i$, where $\zeta_i$ is the
 curve class dual to $H_i$. At a general point of $C$ attach a copy of
 a curve in the class $\zeta_i$ and attach to that another copy of the
 same curve at a variable point. This one-parameter family has
 positive intersection with $T_i$ and does not intersects any of the
 other divisors occurring in the expression of $D$.\medskip

 $NEF^2(X)$ is simplicial, say generated by classes $\gamma_i$.
 $\mbox{Eff}_2(X)$ is simplicial, generated by Schubert varieties (or
 products of Schubert varieties) dual to the previous cone. Each of
 these surfaces $\Gamma_i^*$ have a pencil of rational curves with a
 base point such that for every element $\zeta$ in the pencil $\zeta
 \cdot H_j \leq 1$. To such a pencil attach a fixed curve with $m$
 marked points in the class $\beta - \zeta$ at a base point. The
 resulting one-parameter family has positive intersection with
 $H_{\gamma_i}$ and does not intersect any of the other divisors in
 the expression of $D$. This concludes the proof. \medskip
\end{proof}


\begin{remark}
  Since the divisors $H_{\gamma}, L_i^j$ and $T_i$ are base-point-free
  and $v$ maps base-point-free divisors to base-point-free divisors,
  the theorem remains true if we replace NEF by base-point-free. It is
  clear that Theorem \ref{ample} specializes to Theorem
  \ref{amplecone} when $X$ is a Grassmannian.
\end{remark}




\section{The effective cone}\label{pfmain}

In this section we prove that a divisor lies in the effective cone of
the Kontsevich moduli space $\Kgnb{0,0}(G(k,k+r),d)$ if and only if it
is a non-negative linear combination of the divisors $D_{\mbox{deg}}$,
$D_{\mbox{unb}}$ and the boundary divisors. \medskip

\begin{proof}[Proof of Theorem \ref{main}]

  Since $D_{\mbox{deg}}$, $D_{\mbox{unb}}$ and the boundary divisors
  are effective, any non-negative linear combination of these divisors
  lie in the effective cone. We have to show the converse. \medskip

Note that the divisors $D_{\mbox{deg}}$, $D_{\mbox{unb}}$ and the
boundary divisors form a $\QQ$-basis of $\mbox{Pic}
(\Kgnb{0,0}(G(k,k+r,d),d) \otimes \QQ$. This follows from Theorem
\ref{Pic} and Lemmas \ref{deg}, \ref{notdiv} and \ref{lem-3.2}, but will
also follow from the proof we give. Therefore, we can write the class
of any effective divisor $D$ as a linear combination
\begin{equation}\label{eq}
D = a \ D_{\mbox{deg}} + b \ D_{\mbox{unb}} + \sum_{i=1}^{\lfloor d/2
\rfloor} c_i \ \Delta_{i,d-i}.
\end{equation}
\noindent To conclude the proof we need to show that the coefficients
$a, b$ and $c_i$ are non-negative. If for each of the divisors
$D_{\mbox{deg}}$, $D_{\mbox{unb}}$ and $\Delta_{i,d-i}$ we can
construct a moving curve in $\Kgnb{0,0}(G(k,k+d),d)$ that has positive
intersection number with that divisor and zero with the others, the
theorem follows since $D$ has non-negative intersection with any
moving curve.  We will now construct the necessary moving curves in
$\Kgnb{0,0}(G(k,k+r,d),d)$.
\medskip

For each boundary divisor $\Delta_{i,d-i}$, the paper
\cite{coskun:effective} constructs a sequence of curves $C_i(j)$ in
$\Kgnb{0,0}(\PP^d,d)$ with the following properties:

\begin{enumerate}
\item[(i)] The intersection of $C_i(j)$ with $D_{\mbox{deg}}$ is zero.
\item[(ii)]  The intersection of $C_i(j)$ with $\Delta_{k,d-k}$ is zero if $k
  \not= 1$ or $k \not= i$.
\item[(iii)] The intersection of $C_i(j)$ with $\Delta_{i,d-i}$ is $j(d+1) -
  n(i,d)$ where $n(i,d)$ is a fixed constant depending only on $i$ and
  $d$ and not on $j$.
\item[(iv)] The intersection of $C_i(j)$ with $\Delta_{1,d-1}$ is $n(i,d)
\frac{i(i+1)}{2}$.
\end{enumerate}

\medskip

\noindent We will use the curves $C_i(j)$ as the basis of our construction, so
we recall their construction. Consider the blow-up of $\PP^1 \times
\PP^1$ at $$j(d+1) + n(i,d) \frac{(i+2)(i-1)}{2}$$ general points
$p_h$. We denote the proper transform of the two fiber classes by
$F_1$ and $F_2$, respectively, and we denote the exceptional divisor
lying over $p_h$ by $E_h$. The curve $C_i(j)$ in $\Kgnb{0,0}(\PP^d,d)$
is obtained by considering the image of the one-parameter family of
fibers in the class $F_2$ under the linear system $$d \ F_1 + j
\frac{i(i+1)}{2} \ F_2 - \sum_{h=1}^{j(d+1) - n(i,d)} i \ E_h -
\sum_{h=j(d+1) - n(i,d)+1}^{j(d+1) + n(i,d) \frac{(i+2)(i-1)}{2}}
E_h.$$

\noindent {\bf The proof in the case $k$ divides $d$.}  We assume that
$d=kq$. Fix $k$ general $\PP^{q}$s in $\PP^{k+d-1}$. In order to show
that the coefficient $c_i$ is non-negative, we need a sequence of
curves $B$ in $\Kgnb{0,0}(G(k,k+r,d),d)$ that have intersection zero
with $D_{\mbox{deg}}$ and $D_{\mbox{unb}}$ and $$\frac{B \cdot
  \Delta_k}{B \cdot \Delta_i}$$ tends to zero. Suppose $i = k q_i +
r_i$. We then take $r_i$ families of the type $C_{q_i +1}(j)$ in
$\Kgnb{0,0}(\PP^q,q)$ and $k-r_i$ families of the type $C_{q_i}(j)$ in
$\Kgnb{0,0}(\PP^q,q)$.  Taking their join induces a sequence of moving
curves in $\Kgnb{0,0}(G(k,k+r,d),d)$ with the desired properties.
\medskip



Now we show that the coefficient of $D_{\mbox{deg}}$ is non-negative.
Take the image of $\PP^1 \times \PP^1$ under the linear system
$\OO_{\PP^1 \times \PP^1}(1,q)$ and take a general projection of the
surface to $\PP^q$. Fix $k-1$ degree $q$ rational normal curves in
$k-1$ general $\PP^q$s. Fix an isomorphism between the rational normal
curves of degree $q$ and the fibers of degree $q$ of $\PP^1 \times
\PP^1$. Taking the join of the points corresponding under the
isomorphism provides a family of balanced scrolls that never become
reducible or unbalanced. The scrolls become degenerate finitely many
times when the linear span of the fibers of $\PP^1 \times \PP^1$ is
not all of $\PP^q$. By the universal property of $G(k,k+d)$ this
family of scrolls induces a one-parameter family $C$ of curves in
$\Kgnb{0,0}(G(k,k+r,d),d)$. $C$ has intersection number zero with the
boundary and $D_{\mbox{unb}}$ and positive with $D_{\mbox{deg}}$.
Moreover, Lemma \ref{moving} implies that $C$ is a moving curve.
Consequently, in the expression (\ref{eq}) the coefficient of
$D_{\mbox{deg}}$ has to be non-negative.  \medskip

Now we show that the coefficient of $D_{\mbox{unb}}$ is non-negative.
Recall that $k \geq 2$. By Lemma 2.4 of \cite{coskun:jumping} the
dimension of rational scrolls of degree $d$ and dimension $k$ in
$\PP^{k+d-1}$ is $d(d+2k)-3$. Consider the one-parameter family of
scrolls in $\PP^{k+d-1}$ which contain $d+2k+1$ general points and
intersect $2k-3$ general $(d-2)$-dimensional linear spaces. It is easy
to see that there cannot be any reducible scrolls that contain all the
points. Moreover, since the points are in general position none of the
scrolls containing the points can be degenerate. A general such scroll
will be balanced. This family of scrolls induces a curve in
$\Kgnb{0,0}(G(k,k+r,d),d)$. By Lemma \ref{moving} the resulting curve
is moving and has the desired intersection numbers with
$D_{\mbox{deg}}$, $D_{\mbox{unb}}$ and the boundary divisors. \medskip

\noindent {\bf The proof in the case $k$ does not divide $d$.} In this
case the proof is almost identical. To obtain a moving curve in
$\Kgnb{0,0}(G(k,k+r,d),d)$ that has intersection number zero with the
boundary divisors and $\Dunb$, but positive intersection with $\Ddeg$
take a general projection of the embedding of $\PP^1 \times \PP^1$ by
the linear system $\OO_{\PP^1 \times \PP^1}(1,q+1)$ to $\PP^{q+1}$.
Take the joins of degree $q+1$ fibers with a fixed balanced scroll of
dimension $k-1$ and degree $d-q-1$. This induces a moving curve in
$\Kgnb{0,0}(G(k,k+r,d),d)$ with the desired intersection numbers. By
Lemma 2.4 of \cite{coskun:jumping} the dimension of rational scrolls
of degree $d$ and dimension $k$ in $\PP^{k+d-1}$ is $d(d+2k)-3$. To
obtain a moving curve that has intersection numbers zero with the
boundary and $\Ddeg$, but positive with $\Dunb$, consider the
one-parameter family of scrolls in $\PP^{k+d-1}$ that contain $d+2k+1$
general points and intersects $2k-3$ general $(d-2)$-dimensional
linear spaces. It is easy to see that there cannot be any reducible
scrolls that contain all the points. Moreover, since the points are in
general position none of the scrolls containing the points can be
degenerate. This one-parameter family of scrolls induces a moving
curve in $\Kgnb{0,0}(G(k,k+r,d),d)$ with the desired properties.
\medskip

Finally to show that the coefficients of the boundary divisors are
non-negative we use analogues of the families we used in the case $k$
divides $d$. Suppose that $d = kq + r$. Either $(k-r) q \geq d/2$ or
$r(q+1) \geq d/2$. Let us assume the latter. The argument in the
former case is identical replacing $q+1$ by $q$ and $r$ by $k-r$. Let
$i = rq_i + r_i$. Consider the join of $k-r$ fixed general rational
normal curves of degree $q$ with $r_i$ general curves of type $C_{q_i
  + 1}(j)$ in $\kgnb{0,0}(\PP^{q+1}, q+1)$ and $r-r_i$ general curves
of type $C_{q_i}(j)$ in $\kgnb{0,0}(\PP^{q+1}, q+1)$. The resulting
one-parameter family of scrolls induces a curve $B$ in
$\kgnb{0,0}(G(k,k+d),d)$. By construction $B$ has intersection zero
with $\Dunb$ and $D_{\mbox{deg}}$. As $j$ tends to infinity the ratio
$$\frac{B\cdot \Delta_k}{B \cdot \Delta_i}$$ tends to zero for $k
\not= i$. This concludes the proof of the theorem.




\end{proof}



















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