\documentclass[reqno]{amsart} \pagestyle{plain} \usepackage{hyperref} \usepackage{amsfonts} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amscd} \usepackage{graphics} \usepackage{stmaryrd} \usepackage{epic,eepic,latexsym} \usepackage{color} \usepackage{epsfig} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{thm}[theorem]{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{claim}[theorem]{Claim} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}[theorem]{Question} \newtheorem{remark}[theorem]{Remark} \newtheorem{construction}[theorem]{Construction} \newtheorem{notation}[theorem]{Notation} \newtheorem{example}[theorem]{Example} \newtheorem{hypothesis}[theorem]{Hypothesis} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\CC}{\mathbb{C}} \newcommand{\RR}{\mathbb{R}} \newcommand{\AAA}{\mathbb{A}} \newcommand{\PP}{\mathbb{P}} \newcommand{\PPP}{\PP_{\CC}} \newcommand{\mc}{\mathcal} \newcommand{\mf}{\mathfrak} \newcommand{\mb}{\mathbf} \newcommand{\OO}{\mc{O}} \newcommand{\gr}{\mathbb{G}} \newcommand{\SP}{\text{Spec }} \newcommand{\HB}[2]{\text{Hilb}^{#1}_{#2}} \newcommand{\GR}[1]{\text{Grass}_{#1}} \newcommand{\kgn}[1]{\text{M}_{#1}} \newcommand{\Kgn}[1]{\mathcal{M}_{#1}} \newcommand{\kgnb}[1]{\overline{\text{M}}_{#1}} \newcommand{\Kgnb}[1]{\overline{\mathcal{M}}_{#1}} \newcommand{\Kbm}{\overline{\mathcal{M}}} \newcommand{\Kbmo}{\mathcal{M}} \newcommand{\Ddeg}{D_{\mbox{deg}}} \newcommand{\Dunb}{D_{\mbox{unb}}} \newcommand{\mfF}[1]{\mf{F}_{#1}} \newcommand{\mfM}[1]{\mathfrak{M}_{#1}} \newsavebox{\sembox} \newlength{\semwidth} \newlength{\boxwidth} \newcommand{\Sem}[1]{% \sbox{\sembox}{\ensuremath{#1}}% \settowidth{\semwidth}{\usebox{\sembox}}% \sbox{\sembox}{\ensuremath{\left[\usebox{\sembox}\right]}}% \settowidth{\boxwidth}{\usebox{\sembox}}% \addtolength{\boxwidth}{-\semwidth}% \left[\hspace{-0.3\boxwidth}% \usebox{\sembox}% \hspace{-0.3\boxwidth}\right]% } \newsavebox{\semrbox} \newlength{\semrwidth} \newlength{\boxrwidth} \newcommand{\Semr}[1]{% \sbox{\semrbox}{\ensuremath{#1}}% \settowidth{\semrwidth}{\usebox{\semrbox}}% \sbox{\semrbox}{\ensuremath{\left(\usebox{\semrbox}\right)}}% \settowidth{\boxrwidth}{\usebox{\semrbox}}% \addtolength{\boxrwidth}{-\semrwidth}% \left(\hspace{-0.3\boxrwidth}% \usebox{\semrbox}% \hspace{-0.3\boxrwidth}\right)% } \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ドル \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,ドル 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,ドル 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,ドル 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,ドル set $d= kq+r$ for 0ドル < 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,ドル $\Dunb$ 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,ドル a proper, connected, at-worst-nodal curve of arithmetic genus 0ドル,ドル \item[(ii)] $p_1,\dots,p_m,ドル an ordered sequence of distinct, smooth points of $C,ドル \item[(iii)] and $f:C\rightarrow X,ドル 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ドル$ 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),ドル 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},ドル 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ドル \leq i \leq \lfloor d/2 \rfloor,ドル 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,ドル 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ドル \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ドル \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ドル \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,ドル they stabilize for large $n$. \begin{theorem}\label{stability} For integers $k \geq 2$ and $r \geq 2,ドル 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,ドル 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),ドル respectively. A curve class in $F(k-1,k,k+d)$ is determined by specifying two integers $d_1, d_2,ドル 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,ドル 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ドル < 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},ドル respectively. By the universal property of $\PP S,ドル 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,ドル $\Dunb$ 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}},ドル $D_{\mbox{unb}}$ 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}},ドル $D_{\mbox{unb}}$ 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,ドル 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},ドル 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}},ドル 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ドル< r (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ドル< r (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-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 2ドルq$. 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 2ドルq+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,ドル 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,ドル 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,ドル let $r_i$ be the unique integer 0ドル\leq r_i >> \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. $$ Since $U$ is a dense Zariski open, $\pi_*\mc{F} = (0)$ and $R^1\pi_*\mc{F}$ is a torsion sheaf supported on $\Dunb$ and the divisors $\Delta_{i,d-i}$. The determinant of this torsion sheaf is an effective Cartier divisor $R$ supported on $\Dunb$ and the divisors $\Delta_{i,d-i},ドル cf. \cite{detdiv}. Thus $$ R = r_{\mbox{unb}} \Dunb + \sum_{i\leq \lfloor d/2 \rfloor} r_{i,d-i} \Delta_{i,d-i}. $$ If $\Dunb$ is nonempty $r_{\mbox{unb}}$ is the length of the stalk $(R^1\pi_* \mc{F})_s$ at a generic point $s$ of $\Dunb$. Similarly for $r_{i,d-i}$. Because the formal neighborhood of $s$ in $S$ maps smoothly to the versal deformation space of $(C_s,E_s),ドル the length of $(R^1\pi_*\mc{F})_s$ can be computed using the versal deformation space and thus using any other versal triple. In other words, there exist integers $r_{\mbox{unb}}$ and $r_{i,d-i}$ such that the equation above holds for \emph{every} versal triple $(S,C,E)$. This allows us to compute the coefficients $r_{\mbox{unb}}$ and $r_{i,d-i}$ using special versal triples, i.e., using the method of \emph{test families}. Also Riemann-Roch gives $$ 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}. $$ Let $V$ and $W$ be 2ドル$-dimensional vector spaces. Set $S=\PP V \cong \PP^1$. Let $C=\PP V \times \PP W \cong \PP^1 \times \PP^1$ and let $\pi:C\rightarrow S$ be the projection $\text{pr}_V$. The relative dualizing sheaf of $\pi$ is $\text{pr}_W^*\omega_{\PP(W)}$. And the sheaf $\mc{E}_\pi$ is, $$ \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} $$ Because the section $x$ of $\OO(1)$ is nonzero on $D_+(x),ドル the restriction of $G_x$ to $D_+(x)$ is balanced. However, the restriction of $G_x$ to the fiber over $p$ is unbalanced, $\OO(-1)\oplus \OO(1)$. \medskip\noindent Define locally free sheaves $E'$ and $E$ to be $$ E' := G_x \oplus \OO_C^{\oplus (k-2)}, \ \ E := E'\otimes \text{pr}_W^*\OO(d/k). $$ The datum $(S,C,E)$ is versal on $D_+(x)$. We claim it is also versal at $p,ドル i.e., the induced morphism $\zeta$ from the formal neighborhood of $p$ to the versal deformation space is smooth. %% More generally, if $d$ is not 0ドル,ドル %% replace $\mc{E}$ by $\mc{E}\otimes \text{pr}_W^*\OO_{\PP(W)}(d/k)$ to %% get a 1-morphism $\zeta:S\rightarrow \mfF{k,d}$. The versal deformation space is $k\Sem{t},ドル and the Cartier divisor $\Dunb$ over the deformation space is just $(t)$. The pullback $\zeta^{-1}(\Dunb)$ equals $m\{p\}$ for some positive integer $m$. By Lemma ~\ref{lem-ez}, $\zeta$ is smooth if $m = 1$. Clearly $m$ is the same for both the triple $(S,C,E)$ and the triple $(S,C,E')$. Form the sheaf $\mc{T}_S := R^1\pi_*(E'\otimes \mc{E}_\pi),ドル and form the analogous sheaf $\mc{T}_{\text{versal}}$ on $k\Sem{t}$. As above, $\pi_*(E'\otimes \mc{E}_\pi) = (0)$ and $\mc{T}_S$ is a torsion sheaf supported on $p$. Since $\zeta$ is flat and since $R\pi_*(E'\otimes \mc{E}_\pi)$ is of formation compatible with flat base-change, $$ \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)]) $$ which is canonically $W^\vee$. Since this is 2ドル$-dimensional $$ \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. $$ Since 2ドル = \text{length}(\mc{T}_S) \geq 2m,ドル $m = 1,ドル i.e., $(S,C,E)$ is versal. Because $(S,C,E)$ is versal, because all the divisors $\Delta_{i,d-i}$ are empty, and because $\Dunb$ is the single reduced point $p$ $$ 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). $$ The pushforward to $\PP V$ has degree 4ドルk$; $$ 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, $$ where $r_i$ is the unique integer 0ドル\leq r_i < k$ such that $i-r_i$ is divisible by $k$. Denote $$ 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). $$ Consider the triple $(S,\widetilde{C},E)$. It is clearly versal on $\PP V-\{p\}$. To see that it is versal at $p,ドル observe first the versal deformation space of $(\widetilde{C}_p,E|_{\widetilde{C}_p})$ equals the versal deformation space $\widetilde{C}_p,ドル which is $k\Sem{t}$. Let $m$ be the positive integer such that $(t)$ pulls back to $m\{p\}$. As in \cite[p. 81]{DM}, there is an isomorphism of the formal neighborhood of the node of $\widetilde{C}_p$ in $\widetilde{C}$ with the formal scheme $k\Sem{u,v,s}/\langle uv- s^m \rangle,ドル i.e., an $A_{m-1}$-singularity. Since $\widetilde{C}$ is smooth, $m=1,ドル i.e., $(S,C,E)$ is versal. As above, $r_{i,d-i}$ equals the degree of the first Chern class of $-R{\widetilde{\pi}}_*(\text{End}(E')\otimes \mc{E}_{\widetilde{\pi}}),ドル $$ 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). $$ \end{proof} \section{The ample cone} \label{secample} In this section we prove a generalization of Theorem \ref{amplecone} for products of flag varieties. Our proof will be valid in more general contexts; however, for simplicity of exposition we leave generalizations to the reader. \medskip Let $X$ be a product of flag varieties with Picard number $\rho$. There are $\rho$ morphisms from $X$ to Grassmannians $\phi_i: X \rightarrow G_i$. The cone of NEF divisors on $X$ is simplicial and generated by the pull-backs of the ample divisors on $G_i$ by the morphisms $\phi_i$. We denote these divisors by $H_1, \dots, H_{\rho}$. Let $\beta \in H_2(X, \ZZ)$ be a curve class. For our purposes we may and will assume that $\beta \cdot H_i = e_i>0 $ for every 1ドル \leq i \leq \rho$ (otherwise, we can reduce to the problem of understanding the ample cone when the target has smaller Picard number). \medskip The NEF divisors on $X$ pull-back to NEF divisors on $\Kgnb{0,m}(X, \beta)$ via the evaluation morphisms (see \cite{coskun:nef} Lemma 4.1). We thus obtain the NEF divisors $L_i^j = ev_i^*(H_j)$ on $\Kgnb{0,m}(X, \beta)$ for 1ドル \leq i \leq m $ and $ 1 \leq j \leq \rho$. \medskip The tangency divisor $T$ plays an important role in the description of the ample cone. $T$ is the class of the locus of maps for which the inverse image of a fixed hyperplane in the Pl\"ucker embedding of $G(k,n)$ does not consist of $d$ distinct, smooth points of the domain. For this class to make sense we need that $d> 1$. R. Pandharipande in \cite{pandh:qdivisors} computes the class of the tangency divisor when the target is projective space. Since $T$ on $\Kgnb{0,m} (G(k,n), d))$ is the pull-back of this divisor by the Pl\"ucker embedding, the following formula easily follows from Pandharipande's calculation in \cite{pandh:qdivisors} Lemma 2.3.1: $$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, $$ where $K_i$ is the sum of the boundary divisors where the map has degree $i$ on one component. The divisor $T$ is NEF on $\Kgnb{0,m} (G(k,n), d))$ (\cite{coskun:nef} Lemma 4.1). Assuming that $e_i> 1,ドル pulling-back the tangency divisor on $\kgnb{0,m}(G_i,e_i)$ by $\Kgnb{0,m}(\phi_i)$ we obtain NEF divisors $T_1, \dots, T_{\rho}$ on $\Kgnb{0,m}(X, \beta)$. \medskip The numerically effective codimension two cycles on $X$ are also simplicial and give rise to NEF divisors on $\Kgnb{0,m}(X, \beta)$. Given a cycle $\gamma \in NEF^2(X),ドル the divisor $$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))$$ that maps NEF divisors to NEF divisors. This map is constructed and described in \S 2 of \cite{coskun:nef} for the case of $\PP^n$. The proofs of that section carry over to the case of $X$. The map $v$ is induced by the rational map from $\Kgnb{0,m} (X, \beta)$ to $\kgnb{0,m+e_1+ \cdots + e_\rho}/\mathfrak{S}_{e_1} \times \cdots \times \mathfrak{S}_{ e_\rho}$ obtained by intersecting the image of the map with a fixed union of divisors $H_1 \cup \cdots \cup H_{\rho},ドル marking the points of intersection and then forgetting the map and stabilizing. \begin{theorem}\label{ample} The NEF cone of $\Kgnb{0,m} (X, \beta)$ is the product of the image under $v$ of the NEF cone of $\kgnb{0,m+e_1+ \cdots + e_\rho}/\mathfrak{S}_{e_1} \times \cdots \times \mathfrak{S}_{ e_\rho}$ with the cone spanned by non-negative linear combinations of the classes $L_i^j,ドル for 1ドル \leq i \leq m $ and $ 1 \leq j \leq \rho,ドル $H_{\gamma},ドル for NEF codimension two cycles $\gamma$ on $X,ドル and $T_i,ドル for 1ドル \leq i \leq \rho$ and $e_i> 1$. \end{theorem} \begin{proof} The proof is almost identical to the proof of the main theorem in \cite{coskun:nef}. Clearly any divisor $D$ that lies in this cone is NEF. The main theorem of \cite{dragos:divisors} and Proposition 2.5 of \cite{coskun:nef} imply that divisors in this cone span the Picard group. Hence, we can write any NEF divisor as $$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,ドル 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}},ドル $D_{\mbox{unb}}$ and the boundary divisors. \medskip \begin{proof}[Proof of Theorem \ref{main}] Since $D_{\mbox{deg}},ドル $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}},ドル $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}},ドル $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},ドル 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}$$ general points $p_h$. We denote the proper transform of the two fiber classes by $F_1$ and $F_2,ドル respectively, and we denote the exceptional divisor lying over $p_h$ by $E_h$. The curve $C_i(j)$ in $\Kgnb{0,0}(\PP^d,d)$ is obtained by considering the image of the one-parameter family of fibers in the class $F_2$ under the linear system $$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 2ドルk-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}},ドル $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,ドル 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,ドル but positive with $\Dunb,ドル consider the one-parameter family of scrolls in $\PP^{k+d-1}$ that contain $d+2k+1$ general points and intersects 2ドルk-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} \bibliography{grass} \bibliographystyle{math} \end{document}

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