DIVISORS ON THE SPACE OF MAPS TO GRASSMANNIANS

IZZET COSKUN AND JASON STARR

Abstract. In this note we study the divisor theory of the Kontsevich moduli spaces \overline{M}_{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.

Contents

1. Introduction  1
2. Preliminaries  2
3. The stability of the effective cone  5
4. The divisor class calculations  6
4.1. The divisor class Ddeg  6
4.2. The divisor class Dunb when k does not divide d  8
4.3. The divisor class of Dunb when k divides d  10
5. The ample cone  14
6. The effective cone  16
References  18

1. Introduction

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 \overline{M}_{0,0}(G(k, n), d) of genus-zero stable maps to Grassmannians.

The papers [CHS1] and [CHS2] 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 [CHS1] and [CHS2] apply to more general targets, as we illustrate here with the example of Grassmannians. Our first main theorem is:

Theorem 1.1. Let k, n and d be positive integers such that 2 ≤ k < k + 2 ≤ n. Let m be a non-negative integer such that m + d ≥ 3. There is an injective linear map

v : P ic_{ℚ}(\overline{M}_{0,m+d}/𝔖_{d}) → P ic_{ℚ}(\overline{M}_{0,m}(G(k, n), d))

2000 Mathematics Subject Classification. Primary:14D20, 14E99, 14H10.

During the preparation of this article the second author was partially supported by the NSF grant DMS-0353692 and a Sloan Research Fellowship.

that maps NEF (resp. base-point-free) divisors to NEF (resp. base-point-free) divisors. The NEF (resp. base-point-free) cone of \overline{M}_{0,m}(G(k, n), d) is the product of the cones spanned by H_{σ_{2}}, H_{σ_{1,1}}, T, L_{1}, …, L_{m} and the image under v of the NEF (resp. base-point-free) cone of \overline{M}_{0,m+d}/𝔖_{d}.

Here the action of the symmetric group 𝔖_{d} permutes the last d marked points. The divisors H_{σ_{1,1}} and H_{σ_{2}} are the divisors of curves whose image intersects fixed Schubert varieties Σ_{1,1} and Σ_{2}, respectively. The divisor T is the tangency divisor and L_{i} are the pull-backs of the ample generator of P ic(G(k, n)) by the i-th evaluation morphism.

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 \overline{M}_{0,m}(X, β). Our proof of Theorem 1.1 will remain valid in a more general setting—proving, in particular, the corresponding statement for flag varieties.

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 \overline{M}_{0,0}(G(k, n), d) stabilize as long as n ≥ k + d. We then give a complete description of the effective cone of \overline{M}_{0,0}(G(k, k + d), d). Our methods bound the effective cone of \overline{M}_{0,0}(G(k, n), d) when n < k + d; however, we do not have a complete description of the cone in this case.

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:

Theorem 1.2. The effective cone of \overline{M}_{0,0}(G(k, k + d), d) is a simplicial cone generated by Ddeg, Dunb and the boundary divisors.

The organization of the paper is as follows. In §2 we collect the facts that we use about Kontsevich moduli spaces and state our results more precisely. In §3 we prove that the effective cones of \overline{M}_{0,0}(G(k, n), d) stabilize when n ≥ k + d. In §4 we express the divisor classes Ddeg and Dunb in terms of the standard basis of the Picard group of \overline{M}_{0,0}(G(k, k + d), d). In §5 we prove a generalization of Theorem 1.1. In §6 we prove Theorem 1.2.

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

2. Preliminaries

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

Let X be a smooth, complex projective homogeneous variety. Let β ∈ H_{2}(X, ℤ) be the homology class of a curve. The Kontsevich moduli space of m-pointed, genus-zero stable maps \overline{M}_{0,m}(X, β) provides a useful compactification of the space of rational curves on X whose homology class is β. \overline{M}_{0,m}(X, β) is the smooth, proper Deligne-Mumford stack parameterizing the data of

(i) C, a proper, connected, at-worst-nodal curve of arithmetic genus 0, (ii) p_{1}, …, p_{m}, an ordered sequence of distinct, smooth points of C, (iii) and f : C → X, a morphism with f_{∗}[C] = β 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.

By [KP] \overline{M}_{0,m}(X, β) is irreducible.

Oprea in [Opr] proves that the rational complex codimension one classes on \overline{M}_{0,m}(X, β) are tautological. When X is the Grassmannian G(k, n), one can very explicitly describe the Picard group P ic_{ℚ}(\overline{M}_{0,0}(G(k, n), d) as follows. The cohomology ring of the Grassmannian G(k, n) is generated by the Poincaré duals of the classes of Schubert varieties. We use the convention in [GH] to denote Schubert varieties. There are two divisor classes H_{σ_{1,1}} and H_{σ_{2}} on \overline{M}_{0,0}(G(k, n), d) corresponding to the two codimension-two classes σ_{1,1} and σ_{2} on G(k, n). Informally, these are the classes of the divisors that parameterize the maps whose images intersect fixed Schubert varieties Σ_{1,1} and Σ_{2}, respectively. More precisely, let

ev_{i} : \overline{M}_{0,m}(G(k, n), d) → G(k, n)

denote the i-th evaluation morphism and let

π : \overline{M}_{0,1}(G(k, n), d) → \overline{M}_{0,0}(G(k, n), d)

denote the forgetful morphism. H_{σ_{1,1}} and H_{σ_{2}} are defined as follows:

H_{σ_{1,1}} := π_{∗} ev_{1}^{∗}(σ_{1,1}), H_{σ_{2}} := π_{∗} ev_{1}^{∗}(σ_{2}).

In addition to these, for every 1 ≤ i ≤ ⌊d/2⌋, there is a boundary divisor on \overline{M}_{0,0}(G(k, n), d) whose general point parameterizes a map with a reducible domain C_{1} ∪ C_{2} where the map has Plücker degree i on C_{1} and degree d − i on C_{2}, respectively. We will denote the class of the boundary divisor by Δ_{i,d−i}. The theorem of Oprea reduces to the following in the special case of G(k, n).

Theorem 2.1. ([Opr]) Let 2 ≤ k < k + 2 ≤ n and d ≥ 1 be integers. The divisor classes H_{σ_{1,1}}, H_{σ_{2}} and Δ_{i,d−i} for 1 ≤ i ≤ ⌊d/2⌋ generate Pic_{ℚ}(\overline{M}_{0,0}(G(k, n), d)).

Let P_{k,n,d} be the ℚ-vector space spanned by basis elements labelled by H_{σ_{1,1}}, H_{σ_{2}} and Δ_{i,d−i} for 1 ≤ i ≤ ⌊d/2⌋. Pic_{ℚ}(\overline{M}_{0,0}(G(k, n), d)) can be naturally identified with P_{k,n,d}. Let Eff (k, n, d) denote the image of the effective cone under this identification. We can view the effective cones 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 Eff (k, n, d) is that if we fix k and d, they stabilize for large n.

Theorem 2.2. For integers k ≥ 2 and r ≥ 2, the cone Eff (k, k + r, d) is contained in the cone Eff (k, k + r + 1, d). Furthermore, Eff (k, k + r, d) = Eff (k, k + d, d) for every r ≥ d.

In view of Theorem 2.2 it is especially interesting to describe Eff (k, k + d, d). For the rest of the paper we will concentrate on this case. On \overline{M}_{0,0}(G(k, k+d), d) there are two natural effective ℚ-Cartier divisors Ddeg and Dunb. Their classes will play an essential role in the description of the effective cone of \overline{M}_{0,0}(G(k, k + d), d).

Let Ddeg 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 \overline{M}_{0,0}(ℙ^{d}, d) (see §2.2 of [CHS2]). We will calculate the class Ddeg in §4.1.

The definition of Dunb depends on whether k divides d. If k divides d, then a general point in \overline{M}_{0,0}(G(k, k + d), d) parameterizes a map f : ℙ^{1} → 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 Dunb. 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 π_{1} and π_{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 π_{i}.

Consider the Kontsevich moduli space \overline{M}_{0,0}(F (k − 1, k, k + d), (d − k − 1, d)). The second projection π_{2} induces a 1-morphism

\overline{M}_{0,0}(π_{2}) : \overline{M}_{0,0}(F (k − 1, k, k + d), (d − k − 1, d)) → \overline{M}_{0,0}(G(k, k + d), d).

The morphism \overline{M}_{0,0}(π_{2}) is birational onto its image. Since \overline{M}_{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 \overline{M}_{0,0}(G(k, k + d), d). It follows that the image is an irreducible divisor. We denote its class by Dunb.

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

π_{2} : F (k − r, k, k + d) → G(k, k + d)

induces a birational morphism

\overline{M}_{0,0}(π_{2}) : \overline{M}_{0,0}(F (k − r, k, k + d), ((k − r)q, d)) → \overline{M}_{0,0}(G(k, k + d), d).

The first projection

π_{1} : F (k − r, k, k + d) → G(k − r, k + d)

induces a 1-morphism

\overline{M}_{0,0}(π_{1}) : \overline{M}_{0,0}(F (k − r, k, k + d), ((k − r)q, d)) → \overline{M}_{0,0}(G(k − r, k + d), (k − r)q).

Let S denote the tautological bundle of G(k − r, k + d). ℙS has natural projections ϕ_{1} and ϕ_{2} to G(k − r, k + d) and ℙ^{k+d−1}, respectively. By the universal property of ℙS, the evaluation morphism

ev : \overline{M}_{0,1}(G(k − r, k + d), (k − r)q) → G(k − r, k + d)

induces a 1-morphism to ℙS and hence by composing with ϕ_{2} to ℙ^{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 ℙ^{k+d−1} defines a class in \overline{M}_{0,1}(G(k−r, k+d), (k−r)q) whose push-forward by the forgetful morphism is a divisor D on \overline{M}_{0,0}(G(k − r, k + d), (k − r)q). We define Dunb as

Dunb = \overline{M}_{0,0}(π_{2})_{∗} \overline{M}_{0,0}(π_{1})^{∗}D.

With this notation, we can rephrase Theorem 1.2 as follows:

Theorem 2.3. A divisor D on \overline{M}_{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 + ∑_{i=1}^{⌊d/2⌋} c_{i}Δ_{i,d−i}, a, b, c_{i} ≥ 0

of the divisors Ddeg, Dunb and the boundary divisors.

The proof is similar to the proof of the main theorem in [CHS2]. Recall that a moving curve is a reduced, irreducible curve whose deformations cover a Zariski open subset of \overline{M}_{0,0}(G(k, k + d), d). Every moving curve in \overline{M}_{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 · D ≥ 0. Since the divisors Ddeg, Dunb 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 Ddeg, Dunb and the boundary divisors.

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

Lemma 2.4. A reduced, irreducible curve B ⊂ \overline{M}_{0,0}(G(k, k + d), d) is a moving curve if B contains a point parameterizing a map

f : C → 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.

Proof. Let f : C → G(k, k + d) be a map satisfying the assumptions of the lemma. The orbit of this map under the action of ℙGL(k + d) forms a dense subset of \overline{M}_{0,0}(G(k, k + d), d). Consequently, the ℙGL(k + d) orbit of B covers a Zariski open subset of \overline{M}_{0,0}(G(k, k + d), d). Therefore, B is a moving curve. □

3. The stability of the effective cone

In this section we prove that the effective cones of \overline{M}_{0,0}(G(k, n), d) stabilize when n ≥ k + d. This proves Theorem 2.2 stated in §2.

Proof of Theorem 2.2. For the proof it is more convenient to think of G(k, k + r) = 𝔾(k − 1, k + r − 1) as the parameter space for k − 1-dimensional projective linear spaces in ℙ^{k+r−1}. Let p ∈ ℙ^{k+r} be a point. Let U = ℙ^{k+r} − {p} be the complement of the point. Let

π̃ : U → ℙ^{k+r−1}

denote linear projection from the point p. The morphism π̃ induces a corresponding morphism

π : V → 𝔾(k − 1, k + r − 1),

where V denotes the open subscheme of 𝔾(k − 1, k + r) parameterizing ℙ^{k−1} not containing the point p. This in turn induces a smooth 1-morphism

\overline{M}_{0,0}(π, k + r) : \overline{M}_{0,0}(V, d) → \overline{M}_{0,0}(𝔾(k − 1, k + r − 1), d).

Let i : V → 𝔾(k − 1, k + r) be the open immersion of V into 𝔾(k − 1, k + r). The morphism i also induces a 1-morphism between Kontsevich moduli spaces

\overline{M}_{0,0}(i, d) : \overline{M}_{0,0}(V, d) → \overline{M}_{0,0}(𝔾(k − 1, k + r), d).

The complement of the image has codimension r. Since r ≥ 2, the pull-back morphism on the Picard groups

\overline{M}_{0,0}(i, d)^{∗} : Pic_{ℚ}(\overline{M}_{0,0}(𝔾(k − 1, k + r), d)) → Pic_{ℚ}(\overline{M}_{0,0}(V, d))

is an isomorphism. There is a unique homomorphism

h : Pic_{ℚ}(\overline{M}_{0,0}(𝔾(k − 1, k + r − 1), d)) → Pic_{ℚ}(\overline{M}_{0,0}(𝔾(k − 1, k + r), d))

such that

\overline{M}_{0,0}(π, k + r)^{∗} = \overline{M}_{0,0}(i, d)^{∗} ∘ h.

Since \overline{M}_{0,0}(π, k + r) is smooth it pulls-back effective divisors to effective divisors. It follows that Eff (k, k + r, d) ⊂ Eff (k, k + r + 1, d).

Next suppose that r ≥ d. Let D be any effective divisor on \overline{M}_{0,0}(G(k, k + r, d). A general point in the complement of D parameterizes a stable map f : C → G(k, k + r) such that the image lies in a subgrassmannian G(k, k + d). The linear embedding of the corresponding ℙ^{k+d−1} into ℙ^{k+r−1} induces a 1-morphism

\overline{M}_{0,0}(k + d, k + r) : \overline{M}_{0,0}(G(k, k + d, d) → \overline{M}_{0,0}(G(k, k + r, d).

By construction \overline{M}_{0,0}(k+d, k+r)^{∗}([D]) is the class of the effective divisor \overline{M}_{0,0}(k+ d, k + r)^{−1}(D). Therefore, Eff (k, k + d, d) contains Eff (k, k + r, d). Hence, by the previous paragraph the two are equal. This concludes the proof of Theorem 2.2.

□

4. The divisor class calculations

4.1. The divisor class Ddeg. In this section we express the divisor class Ddeg in terms of the standard generators of Pic_{ℚ}(\overline{M}_{0,0}(G(k, k + d), d)). We recall that Ddeg is the class of the divisor that parameterizes stable maps whose image lies in a proper subgrassmannian G(k, k + d − 1).

Lemma 4.1. The class of the divisor Ddeg in \overline{M}_{0,0}(G(k, k+d), d) can be expressed as

Ddeg = \frac{1}{2d} ((1 − d) H_{σ_{1,1}} + (d + 1) H_{σ_{2}} − ∑_{i=1}^{⌊d/2⌋} i(d − i) Δ_{i,d−i}) .

Proof. We determine the class of the divisor Ddeg 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 oneparameter 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 ℙ^{k−2} and form the cone over this one-parameter family of curves. Each cone gives a one-parameter family of ℙ^{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} · H_{σ_{1,1}} = 0, C_{i} · H_{σ_{2}} = i, C_{i} · Ddeg = 0.

The curve C_{i} is contained in the boundary Δ_{i+1,d−i−1}. The intersection numbers with the boundary divisors are as follows:

C_{i} · Δ_{i+1,d−i−1} = −1, C_{i} · Δ_{i,d−i} = 1, C_{i} · Δ_{1,d−1} = i + 1,

provided that i ≥ 1. The intersection of C_{i} with the other boundary divisors is clearly zero.

Next consider the one-parameter family of degree d rational curves containing d+2 general points and intersecting a general line in a fixed ℙ^{d}. Taking a cone over these curves with a fixed general ℙ^{k−2} induces a curve C in \overline{M}_{0,0}(G(k, k + d), d) with the following intersection numbers (see Lemma 2.1 in [CHS2]):

C · H_{σ_{1,1}} = 0, C · H_{σ_{2}} = \frac{d^{2} + d − 2}{2}, C · Δ_{1,d−1} = \frac{(d + 2)(d + 1)}{2}.

The intersection of C with the remaining boundary divisors is zero.

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 ℙ^{k−2} and take the cone of the curves with vertex this ℙ^{k−2}. This one-parameter family of cones induces a oneparameter family of rational curves in G(k, k+d). Denote by B the resulting family in \overline{M}_{0,0}(G(k, k + d), d). We have the intersection numbers

B · H_{σ_{1,1}} = 0, B · H_{σ_{2}} = 1, B · Ddeg = 1.

B is disjoint from all the boundary divisors Δ_{i,d−i} for i > 1. It is contained in Δ_{1,d−1} with intersection number B · Δ_{1,d−1} = −1. These test curves suffice to determine the class of Ddeg except for the coefficient of H_{σ_{1,1}}.

In order to determine the coefficient of H_{σ_{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 \overline{M}_{0,0}(G(k, k + d), d). A has the following intersection numbers:

A · H_{σ_{1,1}} = d, A · H_{σ_{2}} = d, A · Ddeg = 1.

A is disjoint from all the boundary divisors. The class of Ddeg follows form these intersection numbers. □

4.2. The divisor class Dunb when k does not divide d. In this subsection we express the class of the divisor Dunb in \overline{M}_{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 \overline{M}_{0,0}(G(k, k + d), d) sweep out a k-dimensional scroll of degree d in ℙ^{k+d−1}. Such a scroll has a unique k − r-dimensional minimal subscroll of degree (k − r)q. Recall from §2 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

\overline{M}_{0,0}(π_{2}) : \overline{M}_{0,0}(F (k − r, k, k + d), ((k − r)q, d)) → \overline{M}_{0,0}(G(k, k + d), d).

Note that Dunb may be defined in \overline{M}_{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.

Lemma 4.2. 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 ≤ r_{i} < k. Then the divisor class Dunb in \overline{M}_{0,0}(G(k, n), d) is given by

Dunb = −\frac{(q + 1)(d − q − 1)}{2d}H_{σ_{2}} + \frac{(q + 1)(d + q + 1)}{2d} + ∑_{i=1}^{⌊d/2⌋} γ_{i}Δ_{i}

where

γ_{i} = \begin{cases}\binom{q_{i}+1}{2}(r − k) + \frac{q+1}{2d} (i^{2}(q + 1) − idq_{i} − dr_{i}(q_{i} + 1)) if r_{i} ≤ r \\ \binom{q_{i}+1}{2}(r − k) − (q_{i} + 1)(r_{i} − r) + \frac{q+1}{2d} (i^{2}(q + 1) − idq_{i} − dr_{i}(q_{i} + 1)) \end{cases}

if r < r_{i}.

Proof. We will prove the lemma by intersecting both sides of the equation by oneparameter 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 \overline{M}_{0,0}(G(k, n), d) induced by the family of scrolls. The following intersection numbers are easy to compute

C_{q} · Dunb = q + 1, C_{q} · H_{σ_{2}} = d + q, C · H_{σ_{1,1}} = d − q.

C_{q} clearly has intersection number zero with all the boundary components.

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 2dimensional subscroll of degree 2q + 2. Denote by C_{q+1} the one-parameter family in \overline{M}_{0,0}(G(k, n), d) induced by the family of scrolls. The following intersection numbers are easy to compute:

C_{q+1} · Dunb = 0, C_{q} · H_{σ_{2}} = d + q + 1, C · H_{σ_{1,1}} = d − q − 1.

The intersection of C_{q+1} with the boundary divisors is zero. The coefficients of H_{σ_{2}} and H_{σ_{1,1}} follow from these two families.

Let X be the blow-up of ℙ^{1} × ℙ^{1} at one point. Denote by F_{1} and F_{2} the pullbacks of the two fiber classes from ℙ^{1} × ℙ^{1}. Let E be the exceptional divisor. Let V be the following vector bundle of rank k on X given by

V = \begin{cases}⊕_{j=1}^{k−r}O_{X}(−qF_{1} − mF_{2} + q_{i}E) ⊕ ⊕_{j=1}^{r−r_{i}}O_{X}(−(q + 1)F_{1} − mF_{2} + q_{i}E) \\ ⊕ ⊕_{j=1}^{r_{i}} O_{X}(−(q + 1)F_{1} − mF_{2} + (q_{i} + 1)E) if r ≥ r_{i} \\ ⊕_{j=1}^{k−r_{i}+r}O_{X}(−qF_{1} − mF_{2} + q_{i}E) ⊕ ⊕_{j=1}^{r_{i}−r}O_{X}(−qF_{1} − mF_{2} + (q_{i} + 1)E) \\ ⊕ ⊕_{j=1}^{r} O_{X}(−(q + 1)F_{1} − mF_{2} + (q_{i} + 1)E) if r < r_{i} \end{cases}

Let π : X → ℙ^{1} denote the natural map to ℙ^{1} × ℙ^{1} followed by the first projection. Mapping ℙ(V) to projective space under O_{ℙ(V)}(1) gives a one-parameter family of scrolls of degree d parameterized by ℙ^{1} which in turn induces a one-parameter family B_{i} in \overline{M}_{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} · H_{σ_{2}} = c_{1}^{2}(V) − c_{2}(V), B_{i} · H_{σ_{1,1}} = c_{2}(V).

The Chern classes of V are determined by the Whitney sum formula to be

c_{1}(V) = −dF_{1} − kmF_{2} + iE
c_{1}(V)^{2} = 2kmd − i^{2}
c_{2}(V) = qm(k − r)(k − r − 1) + (q + 1)mr(r − 1) + (2q + 1)mr(k − r)
−q_{i}^{2}\binom{k−r_{i}}{2} − q_{i}(q_{i} + 1)r_{i}(k − r_{i}) − (q_{i} + 1)^{2}\binom{r_{i}}{2}
= md(k − 1) − \frac{1}{2}(q_{i}i(k − 1) + r_{i}(i − q_{i} − 1)).

Hence, we conclude that

B_{i} · H_{σ_{2}} = md(k + 1) − i^{2} + \frac{1}{2}(q_{i}i(k − 1) + r_{i}(i − q_{i} − 1))

and

B_{i} · H_{σ_{1,1}} = md(k − 1) − \frac{1}{2}(q_{i}i(k − 1) + r_{i}(i − q_{i} − 1)).

The intersection number B_{i} · Δ_{j} = δ_{i}^{j}, where δ_{i}^{j} is the Krönecker 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} · Dunb is given by c_{1}(π_{∗}(W^{∗})) The latter may be computed by Riemann-Roch

B_{i} · Dunb = \begin{cases}((q + 1)m − \binom{q_{i}+1}{2})(k − r) if r ≥ r_{i} \\ ((q + 1)m − \binom{q_{i}+1}{2})(k − r) − (q_{i} + 1)(r_{i} − r) if r < r_{i} \end{cases}

The lemma follows by elementary algebra. □

4.3. The divisor class of Dunb when k divides d. In this section we express the class of Dunb in terms of the standard generators of the Picard group of \overline{M}_{0,0}(G(k, k + d), d) when k divides d.

Lemma 4.3. Suppose k divides d. For each integer i, let r_{i} be the unique integer 0 ≤ r_{i} < k such that k divides i − r_{i}. The class of the divisor Dunb satisfies the equation

Dunb = \frac{1}{2k} ((k + 1)H_{σ_{1,1}} + (1 − k)H_{σ_{2}} − ∑_{i=1}^{⌊d/2⌋} r_{i}(k − r_{i})Δ_{i,d−i}) .

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

Definition 4.4. A versal triple is a triple (S, C, E) of (i) a Deligne-Mumford stack S with quasi-projective coarse moduli space, (ii) a family of prestable, genus 0 curves π : C → S, (iii) and a locally free sheaf E on C of rank k and degree d

which is 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.

Lemma 4.5. 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.

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. □

For every versal triple (S, C, E), S contains a collection of reduced, effective Cartier divisors Dunb and (Δ_{i,d−i})_{i∈ℤ} with the following properties.

(i) A codimension 1 point s is in Dunb if and only if

C_{s} ≅ ℙ^{1}, E|_{C_{s}} ≅ O((d/k) − 1) ⊕ O(d/k)^{⊕(k−2)} ⊕ O((d/k) + 1).

(ii) A codimension 1 point s is in Δ_{i,d−i} if and only if

C_{s} = C′_{s} ∪ C′′_{s}, deg(E|_{C′_{s}}) = i, deg(E|_{C′′_{s}}) = d − i.

Denote by U the complement of the divisor Dunb and all the divisors Δ_{i,d−i}. For every geometric point s of U

C_{s} ≅ ℙ^{1}, E|_{C_{s}} ≅ O(d/k)^{⊕k}.

Lemma 4.6. For every versal triple (S, C, E) there is a linear equivalence of ℚ-Cartier divisor classes on S,

2k[Dunb] ∼ π_{∗}C_{2}(End(E)) − ∑_{i≤⌊d/2⌋} r_{i}(k − r_{i})[Δ_{i,d−i}].

First we deduce Lemma 4.3 assuming Lemma 4.6.

Proof of Lemma 4.3. Form the triple with S = \overline{M}_{0,0}(G(k, k + d), d), with π : C → \overline{M}_{0,0}(G(k, k + d), d) the universal family of curves, and with E = f^{∗}S_{k}^{∨}. Here S_{k} is the tautological rank k subbundle on G(k + d, d) and f : C → G(k, k + d) is the universal map.

The stack S is smooth, in fact smooth over the stack 𝔐_{0,0} of prestable, genus 0 curves, cf. [Beh]. Using the relative smoothness, Lemma 4.5 says (S, C, E) is versal if each relative Zariski tangent space of S over 𝔐_{0,0} surjects to the first-order deformation space of E_{s}.

The Euler sequence on G(k, n) is,

0 ⟶ Hom(S_{k}, S_{k}) ⟶ Hom(S_{k}, O_{G(k,n)}^{⊕n}) ⟶ T_{G(k,n)} ⟶ 0.

Pulling back by f and pushing forward by π gives an exact sequence

π_{∗}f^{∗}T_{G(k,n)} ⟶ R^{1}π_{∗}Hom(E, E) ⟶ R^{1}π_{∗}(f^{∗}S_{k}^{∨})^{⊕n}.

The first term is the relative tangent bundle of \overline{M}_{0,0}(G(k, k +d), d) over 𝔐_{0,0}. The second term is the bundle of first-order deformation spaces of E. Since f^{∗}S_{k}^{∨} is globally generated, R^{1}π_{∗}(f^{∗}S_{k}^{∨}) = (0) by [HS, Lemma 2.3(i)]. Therefore (S, C, E) is versal.

Clearly the divisors Dunb and (Δ_{i,d−i})_{i∈ℤ} associated to (S, C, E) are the same as the divisors defined earlier. Finally, by straightforward computation

c_{2}(End(S_{k})) = (k + 1)σ_{1,1} + (1 − k)σ_{2}.
□

Proof of Lemma 4.6. Let

0 ⟶ ω_{π} ⟶ E_{π} ⟶ O_{C} ⟶ 0

be the extension whose class in H^{1}(C, ω_{π}) defines the trace. Form the locally free sheaf F = End(E) ⊗ E_{π}. For every geometric point s of U

F|_{C_{s}} = (O(−1) ⊕ O(−1)) ⊗ End(O(d/k)^{⊕k}) ≅ O(−1)^{⊕2k^{2}}.

Since O(−1) has no cohomology

h^{0}(C_{s}, F|_{C_{s}}) = h^{1}(C_{s}, F|_{C_{s}}) = 0.

Since U is a dense Zariski open, π_{∗}F = (0) and R^{1}π_{∗}F is a torsion sheaf supported on Dunb and the divisors Δ_{i,d−i}. The determinant of this torsion sheaf is an effective Cartier divisor R supported on Dunb and the divisors Δ_{i,d−i}, cf. [KM]. Thus

R = runbDunb + ∑_{i≤⌊d/2⌋} r_{i,d−i}Δ_{i,d−i}.

If Dunb is nonempty runb is the length of the stalk (R^{1}π_{∗}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}π_{∗}F)_{s} can be computed using the versal deformation space and thus using any other versal triple. In other words, there exist integers runb and r_{i,d−i} such that the equation above holds for every versal triple (S, C, E). This allows us to compute the coefficients runb and r_{i,d−i} using special versal triples, i.e., using the method of test families.

Also Riemann-Roch gives

R ∼ 2π_{∗}c_{2}(End(E)),

cf. [dJS]. This gives a linear equivalence of ℚ-Cartier classes

2π_{∗}c_{2}(End(E)) ∼ runbDunb + ∑_{i≤⌊d/2⌋} r_{i,d−i}Δ_{i,d−i}.

Let V and W be 2-dimensional vector spaces. Set S = ℙV ≅ ℙ^{1}. Let C = ℙV × ℙW ≅ ℙ^{1} × ℙ^{1} and let π : C → S be the projection pr_{V}.

The relative dualizing sheaf of π is pr_{W}^{∗} ω_{ℙ(W)}. And the sheaf E_{π} is,

E_{π} = pr_{W}^{∗} (W^{∨} ⊗_{ℚ} O_{ℙ(W)}(−1)).

1. Computation of runb. There is a natural isomorphism of vector spaces,

H^{1}(ℙV × ℙW, pr_{V}^{∗} O(1) ⊗ pr_{W}^{∗} ω_{ℙW}) ≅
H^{0}(ℙV, O(1)) ⊗ H^{1}(ℙW, ω_{ℙW}) ≅ V^{∨}.

Let x be a nonzero element in V^{∨}. Denote by p the unique point of ℙV where x vanishes. Using the isomorphism

Ext^{1}(pr_{W}^{∗} (O(1)), pr_{V}^{∗} (O(1)) ⊗ pr_{W}^{∗} (ω_{ℙW} (1))) ≅
H^{1}(ℙV × ℙW, pr_{V}^{∗} O(1) ⊗ pr_{W}^{∗} ω_{ℙW}),

the element x determines an extension class

0 ⟶ pr_{V}^{∗} O(1) ⊗ pr_{W}^{∗} (ω_{ℙW} (1)) ⟶ G_{x} ⟶ pr_{W}^{∗} O(1) ⟶ 0.

Because the section x of O(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, O(−1) ⊕ O(1).

Define locally free sheaves E′ and E to be

E′ := G_{x} ⊕ O_{C}^{⊕(k−2)}, E := E′ ⊗ pr_{W}^{∗} O(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 ζ from the formal neighborhood of p to the versal deformation space is smooth. The versal deformation space is k \begin{matrix}[ \\ [ & t & ] \\ ] \end{matrix}, and the Cartier divisor Dunb over the deformation space is just (t). The pullback ζ^{−1}(Dunb) equals m{p} for some positive integer m. By Lemma 4.5, ζ 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 T_{S} := R^{1}π_{∗}(E′ ⊗ E_{π}), and form the analogous sheaf T_{versal} on k \begin{matrix}[ \\ [ & t & ] \\ ] \end{matrix}. As above, π_{∗}(E′ ⊗ E_{π}) = (0) and T_{S} is a torsion sheaf supported on p. Since ζ is flat and since Rπ_{∗}(E′ ⊗ E_{π}) is of formation compatible with flat base-change,

length(T_{S}) = m · length(T_{versal}) ≥ m · dim(T_{versal}/tT_{versal}) = m · dim(T_{S} ⊗ κ(p)).

By cohomology and base change

T_{S} ⊗ κ(p) ≅ H^{1}(ℙW, [ω_{ℙW} (1) ⊕ O_{ℙW}^{⊕(k−2)} ⊕ O_{ℙW} (+1)] ⊗ [W^{∨}(−1)])

which is canonically W^{∨}. Since this is 2-dimensional

length(T_{S}) ≥ 2m.

Since T_{S} is a torsion sheaf, length(T_{S}) = c_{1}(T_{S}). By [dJS, Lemma 4.3]

c_{1}(T_{S}) = −c_{1}(Rπ_{∗}(E′ ⊗ E_{π})) = −π_{∗}(c_{1}(E′)^{2} − 2c_{2}(E′)).

By the Whitney sum formula

c_{t}(E′) = c_{t}(G_{x}) = 1 + pr_{V}^{∗} h_{V} + (pr_{V}^{∗} h_{V}) ∩ (pr_{W}^{∗} h_{W}),

where h_{V} = c_{1}(O_{ℙ(V)}(1)) and h_{W} = c_{1}(O_{ℙ(W)}(1)). Hence

length(T_{S}) = 2π_{∗}((pr_{V}^{∗} h_{V}) ∩ (pr_{W}^{∗} h_{W})) = 2.

Since 2 = length(T_{S}) ≥ 2m, m = 1, i.e., (S, C, E) is versal.

Because (S, C, E) is versal, because all the divisors Δ_{i,d−i} are empty, and because Dunb is the single reduced point p

runb = deg_{S}(R).

By Riemann-Roch, R ∼ 2π_{∗}c_{2}(End(E)). Of course End(E) ≅ End(E′). By straightforward computation,

2c_{2}(End(E′)) = −2(rk(E′) − 1)c_{1}(E′)^{2} + 4rk(E′)c_{2}(E′).

Using the computation above, this is,

−2(k − 1)pr_{V}^{∗} (h_{V} ∩ h_{V}) + 4k(pr_{V}^{∗} h_{V}) ∩ (pr_{W}^{∗} h_{W}) = 4k(pr_{V}^{∗} h_{V}) ∩ (pr_{W}^{∗} h_{W}).

The pushforward to ℙV has degree 4k;

runb = 4k.

2. Computation of r_{i,d−i}. Fix a point p in ℙV and a point p′ in ℙW. Let ν : C̃ → C be the blowing-up of ℙV × ℙW along {(p, p′)}. Denote by A the exceptional divisor of ν. Denote by π̃ : C̃ → S the composition pr_{V} ∘ ν. Let i ≤ ⌊d/2⌋ be an integer. Write,

i = kq_{i} + r_{i},

where r_{i} is the unique integer 0 ≤ r_{i} < k such that i − r_{i} is divisible by k. Denote

E′ := OC̃(−(q_{i} + 1)A)^{⊕r_{i}} ⊕ OC̃(−q_{i}A)^{⊕(k−r_{i})}, E := E′ ⊗ ν^{∗}pr_{W}^{∗} O(d/k).

Consider the triple (S, C̃, E).

It is clearly versal on ℙV −{p}. To see that it is versal at p, observe first the versal deformation space of (C̃_{p}, E|C̃_{p}) equals the versal deformation space C̃_{p}, which is k \begin{matrix}[ \\ [ & t & ] \\ ] \end{matrix}. Let m be the positive integer such that (t) pulls back to m\{p\}. As in [DM, p. 81], there is an isomorphism of the formal neighborhood of the node of C̃_{p} in C̃ with the formal scheme k \begin{matrix}[ \\ [ & u, v, s & ] \\ ] \end{matrix} /⟨uv − s^{m}⟩, i.e., an A_{m−1}-singularity. Since C̃ 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 −Rπ̃_{∗}(End(E′)⊗Eπ̃),

r_{i,d−i} = 2π̃_{∗}c_{2}(End(E′)).

Of course

End(E′) = (E′)^{∨} ⊗ E′ = O⊕r_{i}^{2} ; C̃ ⊕ OC̃(−A)^{⊕r_{i}(k−r_{i})} ⊕ OC̃(+A)^{⊕r_{i}(k−r_{i})} ⊕ O⊕(k−r_{i})^{2} ; C̃.

By the Whitney sum formula

c_{t}(End(E′)) = (1 − [A] ∩ [A])^{r(i)(k−r(i))}.

Therefore

r_{i,d−i} = −2deg_{S}π̃_{∗}(r_{i}(k − r_{i})[E] ∩ [E]) = 2r_{i}(k − r_{i})deg_{S}({p}) = 2r_{i}(k − r_{i}).

□

5. The ample cone

In this section we prove a generalization of Theorem 1.1 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.

Let X be a product of flag varieties with Picard number ρ. There are ρ morphisms from X to Grassmannians ϕ_{i} : X → 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 ϕ_{i}. We denote these divisors by H_{1}, …, H_{ρ}. Let β ∈ H_{2}(X, ℤ) be a curve class. For our purposes we may and will assume that β ·H_{i} = e_{i} > 0 for every 1 ≤ i ≤ ρ (otherwise, we can reduce to the problem of understanding the ample cone when the target has smaller Picard number).

The NEF divisors on X pull-back to NEF divisors on \overline{M}_{0,m}(X, β) via the evaluation morphisms (see [CHS1] Lemma 4.1). We thus obtain the NEF divisors L_{i}^{j} = ev_{i}^{∗}(H_{j}) on \overline{M}_{0,m}(X, β) for 1 ≤ i ≤ m and 1 ≤ j ≤ ρ.

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 Plücker 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 [Pa] computes the class of the tangency divisor when the target is projective space. Since T on \overline{M}_{0,m}(G(k, n), d)) is the pull-back of this divisor by the Plücker embedding, the following formula easily follows from Pandharipande’s calculation in [Pa] Lemma 2.3.1:

T = \frac{d − 1}{d}H_{σ_{1,1}} + \frac{d − 1}{d}H_{σ_{2}} + ∑_{i=1}^{⌊d/2⌋} \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 \overline{M}_{0,m}(G(k, n), d)) ([CHS1] Lemma 4.1). Assuming that e_{i} > 1, pulling-back the tangency divisor on \overline{M}_{0,m}(G_{i}, e_{i}) by \overline{M}_{0,m}(ϕ_{i}) we obtain NEF divisors T_{1}, …, T_{ρ} on \overline{M}_{0,m}(X, β).

The numerically effective codimension two cycles on X are also simplicial and give rise to NEF divisors on \overline{M}_{0,m}(X, β). Given a cycle γ ∈ NEF^{2}(X), the divisor

H_{γ} := π_{m+1∗}(ev_{m+1}^{∗}(γ))

is NEF ([CHS1] Lemma 4.1), where ev_{m+1} is the m + 1-st evaluation morphism from \overline{M}_{0,m+1}(X, β) to X and π_{m+1} is the forgetful morphism.

Let \overline{M}_{0,m+e_{1}+···+e_{ρ}}/𝔖_{e_{1}} × ⋯ × 𝔖_{e_{ρ}} denote the quotient of the moduli space of genus-zero stable curves with m + e_{1} + ⋯ + e_{ρ} marked points by the action of the products of the symmetric groups on e_{1}, …, e_{ρ} letters acting by the obvious permutation action. There is a map

v : Pic_{ℚ}(\overline{M}_{0,m+e_{1}+···+e_{ρ}}/𝔖_{e_{1}} × ⋯ × 𝔖_{e_{ρ}}) → Pic_{ℚ}(\overline{M}_{0,m}(X, β))

that maps NEF divisors to NEF divisors. This map is constructed and described in §2 of [CHS1] for the case of ℙ^{n}. The proofs of that section carry over to the case of X. The map v is induced by the rational map from \overline{M}_{0,m}(X, β) to \overline{M}_{0,m+e_{1}+···+e_{ρ}}/𝔖_{e_{1}} × ⋯ × 𝔖_{e_{ρ}} obtained by intersecting the image of the map with a fixed union of divisors H_{1} ∪ ⋯ ∪ H_{ρ}, marking the points of intersection and then forgetting the map and stabilizing.

Theorem 5.1. The NEF cone of \overline{M}_{0,m}(X, β) is the product of the image under v of the NEF cone of \overline{M}_{0,m+e_{1}+···+e_{ρ}}/𝔖_{e_{1}} × ⋯ × 𝔖_{e_{ρ}} with the cone spanned by non-negative linear combinations of the classes L_{i}^{j}, for 1 ≤ i ≤ m and 1 ≤ j ≤ ρ, H_{γ}, for NEF codimension two cycles γ on X, and T_{i}, for 1 ≤ i ≤ ρ and e_{i} > 1.

Proof. The proof is almost identical to the proof of the main theorem in [CHS1]. Clearly any divisor D that lies in this cone is NEF. The main theorem of [Opr] and Proposition 2.5 of [CHS1] imply that divisors in this cone span the Picard group. Hence, we can write any NEF divisor as

D = v(D_{1}) + ∑_{γ∈NEF^{2}(X)} a_{γ}H_{γ} + ∑_{i=1}^{ρ} b_{i}T_{i} + ∑_{1≤i≤ρ,1≤j≤m} c_{i,j}L_{i}^{j}.

We have to show that D_{1} is NEF and the constants a_{γ}, 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 [CHS1].

Briefly take any curve C in \overline{M}_{0,m+e_{1}+···+e_{ρ}}/𝔖_{e_{1}} × ⋯ 𝔖_{e_{ρ}}. 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_{γ} and L_{i}^{j}. Hence D_{1} has to be NEF.

To see that the coefficients of L_{i}^{j} are non-negative take a curve in the class β with two components C_{1} ∪ 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} ≥ 0.

To prove that the coefficients of T_{i} are non-negative consider the following oneparameter family. Take a fixed curve C with m marked points in the class β − 2ζ_{i}, where ζ_{i} is the curve class dual to H_{i}. At a general point of C attach a copy of a curve in the class ζ_{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.

NEF^{2}(X) is simplicial, say generated by classes γ_{i}. Eff_{2}(X) is simplicial, generated by Schubert varieties (or products of Schubert varieties) dual to the previous cone. Each of these surfaces Γ_{i}^{∗} have a pencil of rational curves with a base point such that for every element ζ in the pencil ζ · H_{j} ≤ 1. To such a pencil attach a fixed curve with m marked points in the class β − ζ at a base point. The resulting one-parameter family has positive intersection with H_{γ_{i}} and does not intersect any of the other divisors in the expression of D. This concludes the proof. □

Remark 5.2. Since the divisors H_{γ}, 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 5.1 specializes to Theorem 1.1 when X is a Grassmannian.

6. The effective cone

In this section we prove that a divisor lies in the effective cone of the Kontsevich moduli space \overline{M}_{0,0}(G(k, k + r), d) if and only if it is a non-negative linear combination of the divisors Ddeg, Dunb and the boundary divisors.

Proof of Theorem 2.3. Since Ddeg, Dunb 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.

Note that the divisors Ddeg, Dunb and the boundary divisors form a ℚ-basis of Pic(\overline{M}_{0,0}(G(k, k + r, d), d) ⊗ ℚ. This follows from Theorem 2.1 and Lemmas 4.1, 4.2 and 4.3, but will also follow from the proof we give. Therefore, we can write the class of any effective divisor D as a linear combination

D = a Ddeg + b Dunb + ∑_{i=1}^{⌊d/2⌋} c_{i} Δ_{i,d−i}.  (1)

To conclude the proof we need to show that the coefficients a, b and c_{i} are nonnegative. If for each of the divisors Ddeg, Dunb and Δ_{i,d−i} we can construct a moving curve in \overline{M}_{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 \overline{M}_{0,0}(G(k, k + r, d), d).

For each boundary divisor Δ_{i,d−i}, the paper [CHS2] constructs a sequence of curves C_{i}(j) in \overline{M}_{0,0}(ℙ^{d}, d) with the following properties:

(i) The intersection of C_{i}(j) with Ddeg is zero.

(ii) The intersection of C_{i}(j) with Δ_{k,d−k} is zero if k ≠ 1 or k ≠ i.

(iii) The intersection of C_{i}(j) with Δ_{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. (iv) The intersection of C_{i}(j) with Δ_{1,d−1} is n(i, d)\frac{i(i+1)}{2}.

We will use the curves C_{i}(j) as the basis of our construction, so we recall their construction. Consider the blow-up of ℙ^{1} × ℙ^{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 \overline{M}_{0,0}(ℙ^{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} − ∑_{h=1}^{j(d+1)−n(i,d)} i E_{h} − ∑_{h=j(d+1)−n(i,d)+1}^{j(d+1)+n(i,d) \frac{(i+2)(i−1)}{2}} E_{h}.

The proof in the case k divides d. We assume that d = kq. Fix k general ℙ^{q}s in ℙ^{k+d−1}. In order to show that the coefficient c_{i} is non-negative, we need a sequence of curves B in \overline{M}_{0,0}(G(k, k + r, d), d) that have intersection zero with Ddeg and Dunb and

\frac{B · Δ_{k}}{B · Δ_{i}}

tends to zero. Suppose i = kq_{i} + r_{i}. We then take r_{i} families of the type C_{q_{i}+1}(j) in \overline{M}_{0,0}(ℙ^{q}, q) and k − r_{i} families of the type C_{q_{i}}(j) in \overline{M}_{0,0}(ℙ^{q}, q). Taking their join induces a sequence of moving curves in \overline{M}_{0,0}(G(k, k + r, d), d) with the desired properties.

Now we show that the coefficient of Ddeg is non-negative. Take the image of ℙ^{1} × ℙ^{1} under the linear system O_{ℙ^{1}×ℙ^{1}}(1, q) and take a general projection of the surface to ℙ^{q}. Fix k −1 degree q rational normal curves in k −1 general ℙ^{q}s. Fix an isomorphism between the rational normal curves of degree q and the fibers of degree q of ℙ^{1} × ℙ^{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 ℙ^{1} × ℙ^{1} is not all of ℙ^{q}. By the universal property of G(k, k + d) this family of scrolls induces a one-parameter family C of curves in \overline{M}_{0,0}(G(k, k + r, d), d). C has intersection number zero with the boundary and Dunb and positive with Ddeg. Moreover, Lemma 2.4 implies that C is a moving curve. Consequently, in the expression (1) the coefficient of Ddeg has to be non-negative.

Now we show that the coefficient of Dunb is non-negative. Recall that k ≥ 2. By Lemma 2.4 of [C] the dimension of rational scrolls of degree d and dimension k in ℙ^{k+d−1} is d(d + 2k) − 3. Consider the one-parameter family of scrolls in ℙ^{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 \overline{M}_{0,0}(G(k, k + r, d), d). By Lemma 2.4 the resulting curve is moving and has the desired intersection numbers with Ddeg, Dunb and the boundary divisors.

The proof in the case k does not divide d. In this case the proof is almost identical. To obtain a moving curve in \overline{M}_{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 ℙ^{1} × ℙ^{1} by the linear system O_{ℙ^{1}×ℙ^{1}}(1, q + 1) to ℙ^{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 \overline{M}_{0,0}(G(k, k + r, d), d) with the desired intersection numbers. By Lemma 2.4 of [C] the dimension of rational scrolls of degree d and dimension k in ℙ^{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 ℙ^{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 \overline{M}_{0,0}(G(k, k + r, d), d) with the desired properties.

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 ≥ d/2 or r(q + 1) ≥ 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 \overline{M}_{0,0}(ℙ^{q+1}, q + 1) and r − r_{i} general curves of type C_{q_{i}}(j) in \overline{M}_{0,0}(ℙ^{q+1}, q + 1). The resulting one-parameter family of scrolls induces a curve B in \overline{M}_{0,0}(G(k, k + d), d). By construction B has intersection zero with Dunb and Ddeg. As j tends to infinity the ratio

\frac{B · Δ_{k}}{B · Δ_{i}}

tends to zero for k ≠ i. This concludes the proof of the theorem.

□

References

[Beh] K. Behrend. Gromov-Witten invariants in algebraic geometry. Invent. Math. 127(1997), 601–617.

[C] I. Coskun. Gromov-Witten invariants of jumping curves. to appear in Trans. Amer. Math. Soc.

[CHS1] I. Coskun, J. Harris, and J. Starr. The ample cone of the Kontsevich moduli space. submitted.

[CHS2] I. Coskun, J. Harris, and J. Starr. The effective cone of the Kontsevich moduli space. submitted.

[dJS] A. J. de Jong and J. Starr. Divisor classes and the virtual canonical bundle. preprint, 2005.

[DM] P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. (1969), 75–109.

[GH] P. Griffiths and J. Harris. Principles of Algebraic Geometry. Wiley Interscience, 1978.

[HS] J. Harris and J. Starr. Rational curves on hypersurfaces of low degree. II. Compos. Math. 141(2005), 35–92.

[KP] B. Kim and R. Pandharipande. The connectedness of the moduli space of maps to homogeneous spaces. In Symplectic geometry and mirror symmetry (Seoul, 2000), pages 187–201. World Sci. Publishing, River Edge, NJ, 2001.

[KM] F. F. Knudsen and D. Mumford. The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”. Math. Scand. 39(1976), 19–55.

[Opr] D. Oprea. Divisors on the moduli spaces of stable maps to flag varieties and reconstruction. preprint, math.AG/0404285.

[Pa] R. Pandharipande. Intersections of Q-divisors on Kontsevich’s moduli space \overline{M}_{0,n}(P^{r}, d) and enumerative geometry. Trans. Amer. Math. Soc. 351(1999), 1481–1505.

Department of Mathematics, M.I.T., Cambridge, MA 02139

   E-mail address: coskun@math.mit.edu

Department of Mathematics, M.I.T., Cambridge, MA 02139

   E-mail address: jstarr@math.mit.edu