THE MORI CONES OF MODULI SPACES OF POINTED CURVES OF SMALL GENUS

GAVRIL FARKAS AND ANGELA GIBNEY

1. Introduction

In this paper we describe the Mori cone of curves of the moduli space \overline{M}_{g,n} of npointed stable curves of small genus g. Although important aspects of the birational geometry of \overline{M}_{g,n} heavily depend on whether g is large with respect to n (e.g. the Kodaira dimension), it turns out that surprisingly the problem of determining the cone of curves \overline{NE}_{1}(\overline{M}_{g,n}) can be expressed entirely in terms of the geometry of the moduli spaces \overline{M}_{0,g+n} of rational curves with marked points (cf. [GKM]).

There is a combinatorial stratification of \overline{M}_{g,n} given by topological type and the components of the 1-dimensional stratum, that is, loci of curves with (3g − 4 + n)- nodes are called Faber curves (or F -curves). Our first result is that any curve in \overline{M}_{g,n} is numerically equivalent to an effective combination of these F -curves at least when the genus is relatively small:

Theorem 1. The Mori cone of curves \overline{NE}_{1}(\overline{M}_{g,n}) is generated by F -curves when g ≤ 13, n = 0 or g ≤ 8, n = 1 or g = 6, n = 2.

Thus in this range the cone of nef divisors is described by a simple set of inequalities corresponding to the numerical properties of all F -curves (cf. [GKM]). This result was known when n = 0 for g ≤ 11 (cf. [KMcK]), and when n = 1, g ≤ 6 (cf. [GKM]). The case n = 0, g ≤ 4 was first settled by Faber (cf. [Fa1]).

Already for small n the moduli spaces \overline{M}_{0,n} are quite intricate objects deeply rooted in classical algebraic geometry. For instance \overline{M}_{0,5} is a del Pezzo surface of degree 5 while \overline{M}_{0,6} is a small resolution of two famous modular varieties: the Segre cubic S_{3} ⊆ ℙ^{4} which is the symmetric GIT moduli space of 6 points on ℙ^{1} and is the unique cubic threefold with 10 nodes, and its dual, the Igusa quartic I_{4} ⊆ ℙ^{4} which is the GIT moduli space of 6 points on ℙ^{2} which lie on a conic and whose singular locus consists of 15 double lines (cf. [H]).

Since the space \overline{M}_{0,n} has a combinatorial description somewhat similar to that of a toric variety (although \overline{M}_{0,n} itself is certainly not a toric variety), Fulton asked whether any effective (nef) divisor on \overline{M}_{0,n} is linearly equivalent to an effective combination of boundary divisors corresponding to singular curves. We prove the following result:

Theorem 2. The cone of nef divisors of \overline{M}_{0,6} is contained in the convex hull of boundary classes and it has a natural decomposition into 11 subcones.

The precise inequalities defining these subcones can be found in Section 4. The explicit decomposition of the nef cone of \overline{M}_{0,6} enables us to classify all fibrations of \overline{M}_{0,6}. Recall that a morphism f : X → Y with X and Y being irreducible projective varieties, is said to be a fibration if dim(X) > dim(Y) and f_{∗}O_{X} = O_{Y}, that is, f is its own Stein factorization. We have the following:

Theorem 3. 1. Any fibration of \overline{M}_{0,5} factors through a projection \overline{M}_{0,5} → \overline{M}_{0,4} dropping one of the marked points.

2. Any fibration of \overline{M}_{0,6} factors through a projection \overline{M}_{0,6} → \overline{M}_{0,n}, n ∈ \{4, 5\}, dropping one or two points, or through a projection \overline{M}_{0,6} → \overline{M}_{0,4} × \overline{M}_{0,4} obtained by dropping two disjoint pairs of marked points.

The fact that every nef divisor on \overline{M}_{0,6} is linearly equivalent to an effective combination of boundary divisors (that is, the first part of Theorem 2) has been previously checked by Faber (cf. [Fa2]) and by Keel (using the computer program Porta). The salient features of our Theorem 2 are the method of proof which can be applied in more general situations (see Propositions 8 and 9) and the decomposition of the nef cone of \overline{M}_{0,6} into subcones which in particular leads to a classification of all fibrations of \overline{M}_{0,6}. For example the fibrations \overline{M}_{0,6} → \overline{M}_{0,4} × \overline{M}_{0,4} correspond to nef divisors lying in the boundary of two different chambers (see Section 4 for a precise statement). The main idea in the proof of Theorem 2 is to write canonically every divisor D on \overline{M}_{0,6} as a combination of boundary classes with coefficients being intersection numbers of D with precisely determined F -curves. We hope that a similar approach could lead to a solution to Fulton’s conjecture for all spaces \overline{M}_{0,n}.

We note that Theorem 2 gives a new combinatorial proof that \overline{NE}_{1}(\overline{M}_{0,6}) is spanned by F -curves (cf. [KMcK], Theorem 1.2). The original proof used that \overline{M}_{0,n} is a ℚ-Fano variety for n ≤ 6 (which is not the case for any n ≥ 7).

Sean Keel showed that there are effective divisors on \overline{M}_{0,6} which are not expressible as effective combinations of boundary classes (see also [Ve]). For example, if ϕ : \overline{M}_{0,6} → \overline{M}_{3} is the map obtained by identifying three pairs of points on a 6-pointed rational curve and \overline{H} ⊆ \overline{M}_{3} is the locus of hyperelliptic curves then ϕ^{∗}(\overline{H}) is not linearly equivalent to an effective sum of boundary classes. Hassett and Tschinkel recently proved that the effective cone on \overline{M}_{0,6} is spanned by the boundary classes and the pullbacks ϕ^{∗}(\overline{H}) corresponding to all possibilities of identifying three pairs of points (cf. [HT]). In light of their work, Theorem 2 shows that the nef cone of \overline{M}_{0,6} is very small with respect to the effective cone of divisors. Indeed, while we show that any nef divisor may be expressed as an effective sum of the 25 boundary classes, one needs another 15 divisor classes to describe all effective divisors.

Our proof of Theorem 1 makes use of the following ‘bridge’ theorem (cf. [GKM], Theorem 0.3): if ψ : \overline{M}_{0,g+n}/S_{g} → \overline{M}_{g,n} is the map given by attaching elliptic tails to the first g marked points of a (g+n)-pointed rational curve, then a divisor D on \overline{M}_{g,n} is nef if and only if ψ^{∗}(D) is nef and D meets all F -curves on \overline{M}_{g,n} nonnegatively. In other words, to show that \overline{NE}_{1}(\overline{M}_{g,n}) is generated by F -curves it suffices to prove the similar statement on the space \overline{M}_{0,g+n}/S_{g}.

We note that Theorem 3 should be compared to Gibney’s result that for g ≥ 2 any fibration of \overline{M}_{g,n} factors through a projection to some \overline{M}_{g,i} (i < n) dropping some of the marked points (cf. [G],[GKM], Corollary 0.10). Paradoxically, because of the combinatorial complexity of Pic(\overline{M}_{0,n}), the fibration problem is much more difficult in genus 0 than in higher genus!

Acknowledgments: We are grateful to Igor Dolgachev, Bill Fulton and Sean Keel for many interesting discussions related to this project.

2. Generalities on \overline{M}_{0,n}

We record a few facts about the moduli space \overline{M}_{0,n} of stable rational n−pointed curves. For more information about \overline{M}_{0,n} see for example [Kap] or [Ke]. Throughout the paper we work exclusively with ℚ-divisors and all the Picard groups we consider are with rational coefficients.

A vital codimension-k-stratum is a component of the closure of the locus of points in \overline{M}_{0,n} that correspond to curves with k nodes. The boundary of \overline{M}_{0,n} is composed of the vital codimension 1-strata Δ_{S} where S ⊂ {1, …, n} with |S|, |S^{c}| ≥ 2. We always have the identification Δ_{S} = Δ_{S^{c}}. We denote by δ_{S} the linear equivalence class of Δ_{S} in Pic(\overline{M}_{0,n}). An effective 1-cycle that is numerically equivalent to a vital 1-stratum is also known as an F-curve. By an F-divisor we mean a divisor than nonnegatively intersects the F -curves.

We will consider the tautological classes ψ_{i} = c_{1}(𝕃_{i}) for 1 ≤ i ≤ n, where 𝕃_{i} is the line bundle on \overline{M}_{0,n} whose fibre over the moduli point [C, x_{1}, …, x_{n}] is T_{x_{i}}^{∨}(C). Recall also that there exists an ample divisor class

κ_{1} = ∑_{|S|≤n/2^{S⊂\{1,... ,n\}}} \frac{(|S| − 1)(n − |S| − 1)}{(n − 1)}δ_{S}

whose support is the whole boundary of \overline{M}_{0,n} (cf. [AC]).

For each subset S ⊂ {1, …, n} we denote by G_{S} the stabilizer of δ_{S} under the natural action of S_{n} on \overline{M}_{0,n}. Then the G_{S}-invariant divisor classes of the form

δ_{b}^{S,a} := ∑_{B⊂S^{c},|B|=b^{A⊂S,|A|=a}} δ_{A∪B}

generate Pic(\overline{M}_{0,n})^{G_{S}}. We have the following relation between tautological and boundary classes:

Lemma 1. The tautological classes ψ_{i}, for 1 ≤ i ≤ n have the following average expression in terms of G_{{i}}-invariant boundaries:

ψ_{i} ≡ ∑_{j=1}^{n−3} \frac{(n − 1 − j)(n − 2 − j)}{(n − 1)(n − 2)}δ_{j}^{\{i\},1}.

Proof. We use that given two distinct elements q, r ∈ {1, …, n} − {i}, we have that ψ_{i} ≡ ∑_{q,r∉S^{i∈S}} δ_{S} (cf. [AC], Proposition 1.6). We then average all such relations

obtained by varying q and r.

The following average relation between G_{S}-invariant divisor classes will be used throughout the paper:

Proposition 1. Suppose S ⊂ {1, …, n} has s elements. The following relation in Pic(\overline{M}_{0,n}) holds:

δ_{S} = ∑_{0≤b≤n−s−1^{1≤a≤s,(a,b)≠(s,0)}} η_{s,a,b} δ_{b}^{S,a},

where

η_{s,a,b} := \frac{a(b + s − n)(1 + b + a(n − 1) − n + s − s(a + b))}{s(s − 1)(n − s)(n − s − 1)}.

Proof. We use Keel’s relation in Pic(\overline{M}_{0,n}) (cf. [Ke]): given four distinct elements p, q, r, s ∈ {1, …, n} we have that

∑_{r,s∉T^{p,q∈T}} δ_{T} ≡ ∑_{q,s∉T^{p,r∈T}} δ_{T} ≡ ∑_{q,r∉T^{p,s∈T}} δ_{T} .

Having fixed S we write down all possible such relations for which p, q ∈ S and r, s ∉ S. Then we add them together and average.

It is well known that the boundary classes δ_{S} generate Pic(\overline{M}_{0,n}) (cf. [Ke]). The existence of many relations between the δ_{S}’s, hence the absence of a “canonical” basis of Pic(\overline{M}_{0,n}) reflects the combinatorial complexity of \overline{M}_{0,n}. Using Kapranov’s description of \overline{M}_{0,n} as the space obtained from ℙ^{n−3} after a sequence of (n − 4) blow-ups one sees that ψ_{n} and the boundaries δ_{S∪{n}} with S ⊂ {1, …, n − 1} and |S| ≤ n − 4, constitute a basis for Pic(\overline{M}_{0,n}). In particular ρ(\overline{M}_{0,n}) = 2^{n−1} − \binom{n}{2} − 1. However, this basis singles out the n-th marked points and we chose to express all our calculations in a basis which treats all marked points equally:

Lemma 2. For n ≥ 5, the classes {ψ_{i}}_{i=1}^{n} and δ_{S} where |S|, |S^{c}| ≥ 3 form a basis of Pic(\overline{M}_{0,n}).

Proof. We denote by V ⊂ Pic(\overline{M}_{0,n}) the subspace generated by the classes \{ψ_{i}\}_{i=1}^{n} and {δ_{S}}_{|S|,|S^{c}|≥3}. It is enough to show that δ_{xy} ∈ V for all distinct x, y. From Proposition 1 we obtain that (n − 2)(n − 3)δ_{xy} + 2δ_{2}^{xy,0} − (n − 3)δ_{1}^{xy,1} ∈ V (i).

By writing the relation ψ_{i} = ∑_{x,y∉T^{i∈T}} δ_{T} for all i ∈ {x, y}^{c} and averaging we obtain

that (n − 2)δ_{xy} + 2δ_{2}^{xy,0} ∈ V (ii). Finally, by averaging all relations ψ_{x} + ψ_{y} =

∑_{a,b∉T^{x∈T}} δ_{T} + ∑_{a,b∉T^{y∈T}} δ_{T} over all a, b ∈ {x, y}^{c} we obtain that

2\binom{n − 2}{2}δ_{xy} + 2δ_{2}^{xy,0} + \binom{n − 3}{2}δ_{1}^{xy,1} ∈ V (iii).

Clearly (i) − (iii) imply that δ_{xy} ∈ V.

We will often use the following notation:

Definition 1. For a divisor D ≡ ∑_{1≤i≤n} c_{i}ψ_{i} − ∑_{|S|,|S^{c}|≥3} b_{S}δ_{S} on \overline{M}_{0,n} and for a fixed subset T ⊂ {1, …, n}, we set

I^{T} := ∑_{t∈T} c_{t}, O^{T} := ∑_{t∉T} c_{t}, Σ_{i}^{T,j} := ∑_{B⊂T^{c},|B|=i^{A⊂T,|A|=j}} b_{A∪B}.

We also recall that F -curves in \overline{M}_{0,n} correspond to partitions I, J, K, L of \{1, …, n\} into non-empty subsets. For each such partition we have a map ν : \overline{M}_{0,4} → \overline{M}_{0,n} obtained by attaching 1 + |I|, 1 + |J|, 1 + |K| and 1 + |L|-pointed rational curves at each of the four marked points. Every F -curve in \overline{M}_{0,n} is numerically equivalent to such an image ν(\overline{M}_{0,4}) corresponding to a partition (cf. [GKM], Theorem 2.2).

3. The fibrations of \overline{M}_{0,5}

In this section we first show that any F -nef divisor in \overline{M}_{0,5} can be expressed as an effective sum of boundary classes. Although this result can be proved in various ways we present it because it illustrates our technique for giving a natural presentation of any divisor in terms of boundary classes via averaging. Moreover, it enables us to classify the fibrations of \overline{M}_{0,5}.

For a, b ∈ {1, …, 5} we consider the G_{ab}-invariant sum of F -curves C^{ab} := ∑_{i∈\{a,b\}^{c}} Δ_{abi}. We show that any divisor on \overline{M}_{0,5} has a canonical presentation in terms of boundary divisors.

Proposition 2. If D is any divisor in \overline{M}_{0,5} then

D ≡ ∑_{a,b∈\{1,... ,5\}} \frac{1}{6} (C^{ab} · D) δ_{ab}.

In particular any F -divisor is an effective sum of boundary classes.

Proof. We have seen that \{ψ_{i}\}_{i=1}^{5} forms a basis for Pic(\overline{M}_{0,5}). Let D ≡ ∑_{1≤i≤5} c_{i}ψ_{i} be any divisor on \overline{M}_{0,5}. Using the average formula from Lemma 1

ψ_{i} = \frac{1}{2}(∑_{a≠i} δ_{ai}) + \frac{1}{6}(∑_{a,b≠i} δ_{ab})

we can rewrite D as

D ≡ ∑_{a,b∈\{1,... ,5\}} \frac{1}{6}(3 I^{ab} + O^{ab})δ_{ab}.

The coefficient of δ_{ab} is this expression is just \frac{1}{6}(D·C^{ab}) so the conclusion follows.

Remark. If D ≡ ∑_{i=1}^{5} c_{i}ψ_{i} is an F -divisor on \overline{M}_{0,5} we see that D·Δ_{ab} = O^{ab} ≥ 0 for any a, b ∈ {1, …, 5}. Moreover if D · C^{ab} = 3I^{ab} + O^{ab} = 0 then c_{i} = −(c_{a} + c_{b}) ≥ 0, for all i ∈ {a, b}^{c}.

Next we prove that a nontrivial F -divisor on \overline{M}_{0,5} is either big or the pull-back of an ample divisor under the projection π_{i} : \overline{M}_{0,5} → \overline{M}_{0,4} dropping the i-th point.

Proof of Theorem 3, Part 1. Let D ≡ ∑_{i=1}^{5} c_{i}ψ_{i} be a nontrivial F -divisor. We have the following possibilities:

1. D · C^{ab} > 0 for any a, b ∈ {1, …, 5}. Then using Proposition 2 we can write D ≡ aκ_{1} + (Effective), for some a ∈ ℚ_{>0} and since κ_{1} is ample D has to be big so it does not give rise to a fibration. Thus we may assume that D · δ_{ab} = 0 for some a, b ∈ {1, …, 5}, say D · C^{12} = 0 . There are two possibilities:

2. D · C^{1i} > 0 for all i ∈ {1, 2}^{c}. Then c_{3} = c_{4} = c_{5} = c = −(c_{1} + c_{2}) > 0. Moreover D · C^{ab} > 0 for a, b ∈ {1, 2}^{c} and D · C^{2i} > 0 for all i ∈ {1, 2}^{c}. In this case using Proposition 2 the divisor D can be rewritten as a positive combination

D ≡ \frac{5c + 3c_{1} + c_{2}}{6} D_{1} + \frac{5c + c_{1} + 3c_{2}}{6} D_{2},

where D_{1} = ∑_{a,b∈{1,2}^{c}} δ_{ab} + ∑_{a≠1,2} δ_{1a} and D_{2} = ∑_{a,b∈{1,2}^{c}} δ_{ab} + ∑_{a≠1,2} δ_{2a}.

Since D_{2} = π_{1}^{∗}(δ_{23} + δ_{24} + δ_{34}) and D_{1} = π_{2}^{∗}(δ_{13} + δ_{14} + δ_{34}), it follows that D is the pull-back of an ample divisor under the birational map (π_{1}, π_{2}) : \overline{M}_{0,5} → \overline{M}_{0,4}×\overline{M}_{0,4}, hence it is big.

3. There is an i ∈ {1, 2}^{c} such that D · C^{1i} = 0, say D · C^{13} = 0. Then c_{2} = c_{3} = c_{4} = c_{5} = c > 0 and c_{1} = −2c. Proposition 2 gives that D ≡ c ∑_{a,b≠1} δ_{ab} = cπ_{1}^{∗}(δ_{23} + δ_{24} + δ_{25}), which proves our contention. □

4. The nef cone of \overline{M}_{0,6}

In this section we prove Theorems 2 and 3. The main idea is to canonically write every divisor D on \overline{M}_{0,6} as a linear combination of boundary divisors with coefficients being intersection numbers with specific combinations of F -curves.

We first introduce a number of 1-cycles on \overline{M}_{0,6}. Whenever we refer to a 1cycle as being a weighted sum of F -curves we mean that we divide by the number of irreducible components making up the cycle. Let us fix distinct a, b ∈ {1, …, 6}. By C_{1}^{ab} (respectively C_{2}^{ab}) we denote the weighted sum of F -curves of type (3 : 1 : 1 : 1) (resp. (2 : 2 : 1 : 1)) having both points indexed by a and b on the spine. By C_{3}^{ab} we denote the weighted sum of F -curves of type (3 : 1 : 1 : 1) having neither a nor b on the spine, while C_{4}^{ab} is the weighted sum of F -curves of type (2 : 2 : 1 : 1) with a, b on the same tail.

For a, b ∈ {2, …, 6}, let C_{1}^{1ab} be the weighted sum of F -curves of type (2 : 2 : 1 : 1) having exactly one of the points indexed by elements of {1, a, b} on the spine while the remaining two points are on one of the attached tails. By C_{2}^{1ab} (resp. C_{3}^{1ab}) we denote the weighted sum of F -curves of type (3 : 1 : 1 : 1) having only one of the points (resp. two of the points) indexed by elements of {1, a, b} on the spine.

It may be of interest to note that C_{1}^{ab} and C_{2}^{ab} are the only G_{ab}-invariant F - curves (up to rescaling) that properly intersect Δ_{ab}. Similarly C_{1}^{1ab} is the unique G_{1ab}-invariant F -curve that properly intersects Δ_{1ab} and C_{2}^{1ab} and C_{3}^{1ab} are the only G_{1ab}-invariant curves of type (3 : 1 : 1 : 1) that do not intersect Δ_{1ab} at all.

Throughout this section we use the notation from Definition 1. To simplify things we set Σ^{abc} := Σ_{1}^{abc,2}, Σ := Σ^{abc} + δ_{abc} and I + O := I^{abc} + O^{abc} = I^{ab} + O^{ab}, for any a, b, c ∈ {1, …, 6}. The following lemma describes the intersections of the previously introduced curves with any divisor.

Lemma 3. If D ≡ ∑_{1≤i≤6} c_{i}ψ_{i} −∑_{ij∈\{2,... ,6\}} b_{1ij}δ_{1ij} is any divisor on \overline{M}_{0,6}, then for distinct a, b, c ∈ {1, …, 6} we have that

C_{1}^{ab} · D = I^{ab} + \frac{1}{4}O^{ab} + \frac{1}{4}Σ_{1}^{ab,2}, C_{2}^{ab} · D = I^{ab} − \frac{1}{3}Σ_{2}^{ab,1}, C_{3}^{ab} · D = \frac{3}{4}O^{ab} + \frac{1}{4}Σ_{1}^{ab,2},

C_{4}^{ab} · D = \frac{1}{2}O^{ab} − \frac{1}{2}Σ_{1}^{ab,2}, C_{1}^{abc} · D = \frac{1}{3}(I + O) − b_{abc} − \frac{1}{9}Σ^{abc},

C_{2}^{abc} · D = \frac{1}{3}I^{abc} + \frac{2}{3}O^{abc} + \frac{1}{9}Σ^{abc}, C_{3}^{abc} · D = \frac{2}{3}I^{abc} + \frac{1}{3}O^{abc} + \frac{1}{9}Σ^{abc}.

Proof. This follows from standard intersection calculations as explained in for example [HMo] or [Fa1].

The following sufficient criteria for a divisor on \overline{M}_{0,6} to be big will prove useful a number of times:

Lemma 4. Let {i, j, k, l, m, n} be a permutation of {1, …, 6}. Then any effective sum of boundary classes supported on δ_{mn}, δ_{il}, δ_{jl}, δ_{kl}, δ_{mni}, δ_{mnj} and δ_{mnk} is big.

Moreover any effective class supported on δ_{il}, δ_{jm}, δ_{kn} and on all boundaries δ_{abc} except δ_{ijk} is big as well.

Proof. For the first statement it is enough to consider the pullback of the ample class (δ_{mn}, δ_{mn}, δ_{mn}) under the birational map (π_{ij}, π_{jk}, π_{ik}) : \overline{M}_{0,6} → \overline{M}_{0,4} × \overline{M}_{0,4} × \overline{M}_{0,4} whose components forget the marked points (i, j), (j, k) and (i, k) respectively. To prove the second statement we pull back the class (δ_{jm} + δ_{kn}, δ_{il} + δ_{kn}, δ_{il} + δ_{jm}) via the birational map (π_{il}, π_{jm}, π_{kn}) : \overline{M}_{0,6} → \overline{M}_{0,4} × \overline{M}_{0,4} × \overline{M}_{0,4}.

We have the following canonical presentation of any divisor class on \overline{M}_{0,6}:

Proposition 3. Any divisor D on \overline{M}_{0,6} can be written as

D ≡ ∑_{a,b∈\{1,... ,6\}}(\frac{2}{5}(C_{1}^{ab} · D) + \frac{1}{5}(C_{2}^{ab} · D) )δ_{ab} +

+ ∑_{a,b∈\{2,... ,6\}}(\frac{7}{10}(C_{1}^{1ab} · D) + \frac{1}{15}((C_{2}^{1ab} + C_{3}^{1ab}) · D) + \frac{4}{135}Σ^{1ab})δ_{1ab}.

Proof. We perform two canonical modifications of the expression of any divisor on \overline{M}_{0,6} in the basis referred to in Lemma 2. In this way we get two presentations for any divisor on \overline{M}_{0,6}. The expression from Proposition 3 is obtained by taking a suitable linear combination of them. Note that if D is an F -divisor the δ_{ab} part of the expression of D is always effective.

We start with a divisor D ≡ ∑_{1≤i≤6} c_{i}ψ_{i} − ∑_{j,k∈{2,... ,6}} b_{1jk}δ_{1jk}. We replace each ψ_{i} by its average expansion provided by Lemma 1 to get that D is linearly equivalent to

D^{I} = ∑_{j,k∈\{2,... ,6\}}(\frac{3}{10}(I + O) − b_{1jk}) δ_{1jk} + ∑_{a,b∈\{1,... ,6\}}(\frac{3}{5}I^{ab} + \frac{1}{10}O^{ab}))δ_{ab}.

Next, in D^{I} we replace each class δ_{1jk} by its average formula from Proposition 1,

δ_{1jk} = \frac{2}{9} ∑_{b∈\{1,j,k\}^{c}^{a∈\{1,j,k\}}} δ_{ab} − \frac{1}{3} ∑_{ab∈\{1,j,k\}^{c}^{ab∈\{1,j,k\} or}} δ_{ab} + \frac{1}{9} ∑_{a,b≠j,k} δ_{1ab}, (1)

to get that D can also be written as

D^{II} = ∑_{j,k∈\{2,... ,6\}}(\frac{3}{10}(I+O)−\frac{1}{9}Σ^{1jk})δ_{1jk}+ ∑_{a,b∈\{1,... ,6\}}(\frac{3}{5}I^{ab}+\frac{1}{10}O^{ab}+\frac{1}{3}Σ_{1}^{ab,2}−\frac{2}{9}Σ_{2}^{ab,1})δ_{ab}.

We now write that D ≡ \frac{3}{10}(\frac{7}{3}D^{I} + D^{II}) and by using the intersection numbers computed in Lemma 3 we get exactly the desired expression for D.

To simplify notation we shall rewrite the expression from Proposition 3 as

\frac{10}{3}D = \frac{7}{3}D^{I} + D^{II} = ∑_{a,b∈\{1,... ,6\}} ζ_{ab}δ_{ab} + ∑_{a,b∈\{2,... ,6\}} ζ_{1ab}δ_{1ab}. (2)

Thus ζ_{ab} = 2I^{ab} + \frac{1}{3}O^{ab} + \frac{1}{3}Σ_{1}^{ab,2} − \frac{2}{9}Σ_{2}^{ab,1} and ζ_{1ab} = I + O − \frac{20}{9}b_{1ab} − \frac{1}{9}Σ. We have already seen that for an F -divisor all the coefficients ζ_{ab} are ≥ 0. Moreover, in Proposition 4 we prove that at most one of the coefficients ζ_{1ij} can be < 0. If this happens, we replace δ_{1ij} by its average expression (1) spreading the negativity of ζ_{1ij} among all boundary classes. We show that the resulting expression becomes effective thus proving Theorem 2. This procedure gives a decomposition of the nef cone of \overline{M}_{0,6} into 11 natural subcones: one described by inequalities ζ_{1ab} ≥ 0 for all a, b ∈ {2, …, 6} and the remaining 10 given by inequalities ζ_{1ij} ≤ 0 for i, j ∈ {2, …, 6}. More precisely we have the following result:

Proposition 4. Let D be an F -divisor on \overline{M}_{0,6} with ζ_{1ij} < 0 for some i, j ∈ {2, …, 6}. Then D is big and there exists a big effective combination of boundary classes B_{1ij} such that

D ≡ B_{1ij} + ∑_{b∈\{1,i,j\}^{c}^{a∈\{1,i,j\}}}(\frac{1}{6}(C_{1}^{1ij} · D) + \frac{2}{3}(C_{1}^{ab} · D)) δ_{ab} +

+ ∑_{a,b∈\{1,i,j\}^{c}^{a,b∈\{1,i,j\} or}} (\frac{2}{5}(C_{1}^{ab} · D) + \frac{1}{5}(C_{2}^{ab} · D)) δ_{ab} + ∑_{a,b≠i,j} \frac{2}{3}(C_{ab} · D) δ_{1ab} ,

where C_{ab} is an effective sum of F -curves. In particular Theorem 2 follows.

Proof. After replacing δ_{1ij} by its average expression in (2) we obtain the identity (*)

\frac{10}{3}D ≡ ∑_{b∈\{1,i,j\}^{c}^{a∈\{1,i,j\}}}(ζ_{ab} + \frac{2}{9}ζ_{1ij})δ_{ab} + ∑_{a,b∈\{1,i,j\}^{c}^{a,b∈\{1,i,j\} or}}(ζ_{ab} − \frac{1}{3}ζ_{1ij})δ_{ab} + ∑_{a,b≠i,j}(ζ_{1ab} + \frac{1}{9}ζ_{1ij})δ_{1ab}.

We set ρ := −(I + O + Σ). Since 0 > ζ_{1ij} ≥ ζ_{1ij} − \frac{5}{2}C_{1}^{1ij} · D = −\frac{1}{6}ρ, we obtain that ρ > 0. It is rather straightforward to check using Lemma 3 that

ζ_{ab} + \frac{2}{9}ζ_{1ij} = \frac{5}{9}C_{1}^{1ij} · D + \frac{20}{9}C_{1}^{ab} · D + \frac{5}{27}ρ, for a ∈ \{1, i, j\} and b ∈ \{1, i, j\}^{c}, (3)

\frac{1}{9}ζ_{1ij} + ζ_{1ab} = \frac{2ρ − 8ζ_{1ij}}{9} + \frac{20}{9}(I + O − b_{1ij} − b_{1ab}), for \{a, b\} ≠ \{i, j\}, (4)

while obviously ζ_{ab} − \frac{1}{3}ζ_{1ij} > 0 for a, b ∈ \{1, i, j\} or a, b ∈ \{1, i, j\}^{c}. We claim that (*) is already an effective representation of \frac{10}{3}D. As it turns out we can prove a little more than that.

For a, b ∈ {2, …, 6} such that {a, b} ≠ {i, j} we define an effective 1-cycle C_{ab} such that C_{ab} · D = (I + O) − b_{1ij} − b_{1ab}. By passing to the complement if necessary, we may assume that {i, j} ∩ {a, b} = ∅. We denote by k the remaining marked point, hence \{1, …, 6\} = \{1, i, j, k, a, b\}. We then take C_{ab} := \frac{1}{2}(2C_{3}^{1k} + C_{4}^{1k} + C′), where C′ is the F -curve of type (2 : 2 : 1 : 1) with i, j and a, b respectively sitting on different tails.

We define the divisor class B_{1ij} by the formula

B_{1ij} = \frac{3}{10}(\frac{5ρ}{27} ∑_{b∈\{1,i,j\}^{c}^{a∈\{1,i,j\}}} δ_{ab} − \frac{ζ_{1ij}}{3} ∑_{a,b∈\{1,i,j\}^{c}^{a,b∈\{1,i,j\} or}} δ_{ab} + \frac{2ρ − 8ζ_{1ij}}{9} ∑_{a,b≠i,j} δ_{1ab}). (5)

All the coefficients in this expression are positive while the support of B_{1ij} is ∑_{S≠{1,i,j}} Δ_{S}, which is a big divisor (use Lemma 4).

We shall use Proposition 4 to classify all fibrations of \overline{M}_{0,6}. We have already seen that an F -divisor D for which there exists a coefficient ζ_{1ij} < 0, has to be big, hence it does not give rise to a fibration. The divisor D is also big when ζ_{1ab} > 0 and ζ_{ab} > 0 for all a and b (use the existence of the ample class κ_{1}), so we are left with classifying nontrivial F -divisors D for which all of the coefficients in (2) are nonnegative and at least one of them is 0. We have three cases to consider:

1. There are at least two coefficients ζ_{1ab} which are equal to 0, that is, D lies in the intersection of two of the subcones making up the nef cone of \overline{M}_{0,6}. Then we show that D is the pullback of an effective divisor via a fibration \overline{M}_{0,6} → \overline{M}_{0,4} ×\overline{M}_{0,4} obtained by forgetting two disjoint pairs of marked points.

2. There is an i ∈ {1, …, 6} such that ζ_{ij} = 0 for all j ≠ i. Then D is the pullback of an effective divisor via the fibration π_{i} : \overline{M}_{0,6} → \overline{M}_{0,5} forgetting the i-th marked point.

3. If neither of the previous situations occurs then we show that D is big.

The following observation will come into nearly every argument in the rest of this section:

Lemma 5. Let D ≡ ∑_{i=1}^{6} c_{i}ψ_{i} − ∑_{i,j∈{2,... ,6}} b_{1ij}δ_{1ij} be a nontrivial F -divisor on \overline{M}_{0,6} and \{a, b, i, j, m, n\} a permutation of \{1, …, 6\}. If ζ_{ij} = ζ_{ab} = 0 then ζ_{mn} > 0. Moreover, if ζ_{ia} = ζ_{ib} = ζ_{im} = ζ_{in} = 0 then ζ_{ij} = 0 as well.

Proof. Without loss of generality we may assume that ζ_{12} = ζ_{34} = ζ_{56} = 0 and we prove that in this case D is trivial. Our assumption implies that C_{1}^{12} · D = 0, from which we can write that c_{i} + b_{12i} = −(c_{1} + c_{2}), for all i ∈ {1, 2}^{c}. Similarly c_{j} + b_{34j} = −(c_{3} + c_{4}), for all j ∈ {3, 4}^{c} and c_{k} + b_{56k} = −(c_{5} + c_{6}) when k ∈ {5, 6}^{c}. It is easy to see that these relations imply that all the c_{i}’s are equal, that is, c_{i} = c for i ∈ {1, …, 6}, hence b_{12i} = −3c, for each i ∈ {1, 2}^{c}. Similarly b_{34j} = −3c for j ∈ {3, 4}^{c} and b_{56k} = −3c for each k ∈ {5, 6}^{c}.

On the other hand C_{2}^{12}·D = 0 which implies in particular that c_{1}+c_{2}−b_{134}−b_{156} = 0, thus giving that c_{i} = 0 for all i ∈ {1, …, 6}. It immediately follows that the boundary coefficients must vanish too, hence D is trivial.

For the second part, let us assume that ζ_{12} = ⋯ = ζ_{15} = 0 and we prove that ζ_{16} = 0. Since C_{1}^{1i} · D = 0 for all i ∈ {2, …, 5} we have that b_{1i6} = −I^{1i6} for all i ∈ {2, …, 5} which yields C_{1}^{16} · D = 0. We also know that C_{2}^{1i} · D = 0 which turns out to be equivalent with 2c_{1} + I + O = 0. It follows that C_{2}^{16} · D = 0 as well, hence ζ_{16} = 0.

We proceed with the classification of all F -divisors on \overline{M}_{0,6}. The next lemmas deal with the first two situations:

Lemma 6. Let D ≡ ∑_{i=1}^{6} c_{i}ψ_{i} − ∑_{i,j∈\{2,... ,6\}} b_{1ij}δ_{1ij} be an F -divisor on \overline{M}_{0,6} such that two coefficients ζ_{1ij} vanish, say ζ_{1ij} = ζ_{1kl} = 0, where {1, i, j, k, l, m} is a permutation of {1, …, 6}. Then D is the pullback of an effective divisor via the fibration ϕ = (π_{ij}, π_{kl}) : \overline{M}_{0,6} → \overline{M}_{0,4} × \overline{M}_{0,4} whose components forget the marked points labelled (i, j) and (k, l) respectively.

Proof. We use the notation from the proof of Proposition 4. Without loss of generality we may assume that ζ_{123} = ζ_{145} = 0. From (4) we obtain that ρ = −(I+O+Σ) = 0 and that C_{16} ·D = 0 which implies that C_{3}^{16} ·D = C_{4}^{16} ·D = 0. Since ζ_{123} = 0, we can also write that ρ/15 = C_{1}^{123} · D = 0 and similarly C_{1}^{145} · D = 0. Thus the intersection numbers of D with every component of C_{3}^{16}, C_{4}^{16}, C_{1}^{123} and C_{1}^{145} respectively has to be 0. This gives rise to 28 relations between the coefficients c_{i} and b_{1ij}. By writing out these relations it turns out that (2) can be rewritten as D ≡ (α + β)(δ_{124} + δ_{134} + δ_{125} + δ_{135}) + α(δ_{12} + δ_{13} + δ_{62} + δ_{63} + δ_{146} + δ_{156} + δ_{16} + δ_{23})+ +β(δ_{14} + δ_{15} + δ_{64} + δ_{65} + δ_{126} + δ_{136} + δ_{16} + δ_{45}), where α = \frac{2}{3}(c_{1} + c_{2}) ≥ 0 and β = \frac{2}{3}(c_{1} − c_{2}) ≥ 0. In order to finish the proof it is enough to notice that D ≡ α π_{45}^{∗}(δ_{12} + δ_{23} + δ_{13}) + β π_{23}^{∗}(δ_{14} + δ_{45} + δ_{15}).

Remark. Lemma 6 also shows that if D is a nontrivial F -divisor on \overline{M}_{0,6} then at most two of the coefficients ζ_{1ab} can vanish.

Lemma 7. Let D be an F -divisor such that there exists i ∈ {1, …, 6} with ζ_{ij} = 0 for all j ≠ i. Then D is the pullback of an effective divisor under the projection π_{i} : \overline{M}_{0,6} → \overline{M}_{0,5} dropping the i-th marked point.

Proof. Clearly we can assume that i = 1. The hypothesis ζ_{1j} = 0 for j ∈ {2, …, 6} is equivalent to C_{1}^{1j} ·D = C_{2}^{1j} ·D = 0 for all j ∈ {2, …, 6}. This gives that b_{1ij} = −I^{1ij} for all i, j ∈ {2, …, 6}. It also follows that 2c_{1} + I + O = 0 and I + O + Σ = 0. Then (2) reads

D ≡ ∑_{i,j∈\{2,... ,6\}} \frac{2}{3}I^{ij}(δ_{ij} + δ_{1ij}) = π_{1}^{∗}( ∑_{i,j∈\{2,... ,6\}} \frac{2}{3}I^{ij}δ_{ij}).

Proof of Theorem 3, Part 2. We start with a nontrivial F -divisor D on \overline{M}_{0,6} with ζ_{1ab} ≥ 0 for all a, b ∈ {2, …, 6} and such that at most one coefficient ζ_{1ab} is equal to 0, say ζ_{123} = 0. Moreover, we can assume that for each i there is a j ≠ i such that ζ_{ij} ≠ 0. Then we show that D is big. Note that the case ζ_{1ab} > 0 for all a, b ∈ {2, …, 6} is similar (and simpler).

Lemma 5 limits the number of coefficients ζ_{ij} that can vanish and a case by case analysis shows that we can always find sufficiently many boundaries δ_{ij} on which D is supported. Then we apply Lemma 4 to conclude that D is big. □

5. The Mori cone of \overline{M}_{g}

In this section we show that \overline{NE}_{1}(\overline{M}_{g}) is spanned by F -curves for all g ≤ 13. To prove this, it is enough to show that every S_{g}-invariant extremal ray on \overline{M}_{0,g} is generated by an F -curve (cf. [GKM], Theorem 0.3). We achieve this inductively by writing every nontrivial S_{g}-invariant nef divisor on \overline{M}_{0,g} as a sum K_{\overline{M}_{0,g}} + ∑_{S} a_{S}δ_{S}, where 0 ≤ a_{S} ≤ 1 for all S. We also notice that for any g ≥ 14 there are S_{g}-invariant F -divisors on \overline{M}_{0,g} not of this form, thus hinting that the nature of \overline{M}_{g} changes in a subtle way when g = 14. Finally we present a combinatorial set-up enabling us to compute Mori cones of moduli spaces of 1 and 2-pointed curves of genus g ≤ 8. It is clear that in the same way at least a couple of other cases can be settled as well.

We start by setting some notation. We denote by M̃_{0,n} := \overline{M}_{0,n}/S_{n} and we identify divisors on M̃_{0,n} with S_{n}-invariant divisors on \overline{M}_{0,n}. The spaces M̃_{0,n} are interesting for their own sake. For instance M̃_{0,2g+2} is isomorphic to the closure in \overline{M}_{g} of the locus of hyperelliptic curves of genus g. For 2 ≤ i < ⌊n/2⌋, we set

B_{i} := ∑_{|S|=i^{S⊂{1,... ,g}}} δ_{S}. When i = n/2 we define B_{i} := ∑_{|S|=i,1∈S^{S⊂{1,... ,g}}} δ_{S}.

Keel and McKernan proved the following results about the Mori theory of M̃_{0,n} (cf. [KMcK], Theorem 1.3):

Proposition 5. 1. The effective cone \overline{NE}^{1}(M̃_{0,n}) is generated by the classes of

the divisors B_{i} for 2 ≤ i ≤ ⌊n/2⌋. Any nontrivial nef divisor on M̃_{0,n} is big.

2. For n ≤ 7 the cone of curves \overline{NE}_{1}(\overline{M}_{0,n}) is generated by F -curves. We also have that for n ≤ 11 the cone \overline{NE}_{1}(M̃_{0,n}) is spanned by F -curves.

Remark. The previous result combined with Theorem 0.3 from [GKM] gives that \overline{NE}_{1}(\overline{M}_{g,n}) is spanned by F -curves whenever g + n ≤ 7. We also obtain that for g ≤ 11 every F -divisor on \overline{M}_{g} is nef. We shall extend this result for all g ≤ 13.

We recall that for any S ⊂ {1, …, n} such that |S|, |S^{c}| ≥ 2, there is an isomorphism

ϕ : \overline{M}_{0,|S|+1} × \overline{M}_{0,|S^{c}|+1} ⟶ Δ_{S} ⊆ \overline{M}_{0,n}

given by attaching a rational (|S| + 1)-pointed curve to a rational (|S^{c}| + 1)-pointed curve at a point x. It turns out that ϕ induces an isomorphisms between Mori cones \overline{NE}_{1}(Δ_{S}) = \overline{NE}_{1}(\overline{M}_{0,|S|+1}) × \overline{NE}_{1}(\overline{M}_{0,|S^{c}|+1}) (cf. [KMcK], Lemma 3.8). Moreover, if π_{1} : Δ_{S} → \overline{M}_{0,|S|+1} and π_{2} : Δ_{S} → \overline{M}_{0,|S^{c}|+1} are the two projections then

N_{Δ_{S}/\overline{M}_{0,n}} ≡ (π_{1})^{∗}(−ψ_{x}) + (π_{2})^{∗}(−ψ_{x})

(cf. [KMcK], Lemma 4.5). Since the tautological classes ψ_{x} are nef (cf. [Kap]), it follows that Δ_{S} has anti-nef normal bundle, that is, C · Δ_{S} ≤ 0 for every irreducible curve C ⊆ Δ_{S}. We shall often use certain maps between moduli spaces which we call boundary restrictions:

Definition 2. For m, n ≥ 3 and n_{x_{1}}, …, n_{x_{m}} ≥ 1 such that n = n_{x_{1}} + ⋯ + n_{x_{m}}, we define the map ν : \overline{M}_{0,m} ⟶ \overline{M}_{0,n} which takes a rational m-pointed curve (C, x_{1}, …, x_{m}) to a rational n-pointed curve by attaching a fixed rational (n_{x_{i}} + 1)- pointed curve at each point x_{i} for 1 ≤ i ≤ m.

Note that if n_{x_{i}} = 1 for some i then this amounts to not having attached anything at x_{i}. Moreover any composition of boundary restrictions will be homotopic to a single boundary restriction, in particular they will induce the same map in homology. We make the following simple observation:

Proposition 6. Given integers g, n with g + n ≥ 8, to conclude that \overline{NE}_{1}(\overline{M}_{g,n}) is generated by F -curves it suffices to prove that for all F -divisors D on \overline{M}_{0,g+n}/S_{g} and for all boundary restrictions ν : \overline{M}_{0,k} → \overline{M}_{0,g+n}, where 8 ≤ k ≤ g + n, the pullback ν^{∗}(D) is a nonnegative combination of boundary divisors.

Proof. We apply [GKM], Theorem 0.3. We start with an F -divisor D on \overline{M}_{0,g+n}/S_{g} and we want to show that D is nef. Since D is a nonnegative combination of boundary divisors we only have to show that C · D ≥ 0 for all irreducible curves in a boundary divisor Δ_{S} ≅ \overline{M}_{0,|S|+1} × \overline{M}_{0,|S^{c}|+1}. By hypothesis D_{|\overline{M}_{0,|S|+1}} and D_{|\overline{M}_{0,|S^{c}|+1}} are both effective combinations of boundary, hence we have to test the nefness of D only against curves sitting in the boundary of \overline{M}_{0,|S|+1} and of \overline{M}_{0,|S^{c}|+1} and we can descend all the way to a moduli space \overline{M}_{0,n} with n ≤ 7. Since in this range the F -curves generate \overline{NE}_{1}(\overline{M}_{0,n}) (cf. Proposition 5), the conclusion follows.

Let us consider a boundary restriction ν : \overline{M}_{0,m} → \overline{M}_{0,n} given by the partition (n_{x_{1}}, …, n_{x_{m}}) of n where we assume that n_{x_{j}} ≥ 2 ⇔ j ∈ {1, …, r}. We also denote by A := {x_{r+1}, …, x_{m}} the set of remaining marked points, hence n_{y} = 1 for all y ∈ A. For 2 ≤ i ≤ m − 2 and for S ⊂ {x_{1}, …, x_{r}} we define

B_{i}^{S} := ∑_{T ∩{x_{1},... ,x_{r}}=S^{T ⊂{1,... ,m},|T |=i}} δ_{T} .

The adjunction formula for a bounday restriction ν : \overline{M}_{0,m} → \overline{M}_{0,n} reads

ν^{∗}(K_{\overline{M}_{0,n}}) = K_{\overline{M}_{0,m}} + ∑_{n_{x}≥2} ψ_{x}. (6)

The next statement describes the effect a boundary restriction has on homology.

Proposition 7. Let D ≡ ∑_{i=2}^{⌊n/2⌋} r_{i}B_{i} be a divisor on M̃_{0,n}. If ν : \overline{M}_{0,m} → \overline{M}_{0,n} is

a boundary restriction then ν^{∗}D ≡ ∑_{|S|≤⌊r/2⌋,i≥|S|^{S⊂{x_{1},... ,xr}}} c_{i}^{S}B_{i}^{S}, where

c_{i}^{S} := r_{i+∑_{x∈S} n_{x}−|S|} − \frac{(m − i)(m − 1 − i)(∑_{x∈S} r_{n_{x}}) + i(i − 1)(∑_{x∈S^{c}} r_{n_{x}})}{(m − 1)(m − 2)}.

Proof. We start with the case when only one n_{x} is ≥ 2. We obtain that ν^{∗}(B_{i}) = B_{i}^{∅} + B_{i−n_{x}+1}^{x} − (?i)ψ_{i}, where (?i) = 1 if i = n_{x} and 0 otherwise. By iteration, in the case when n_{x_{j}} ≥ 2 for 1 ≤ j ≤ r, we can write that

ν^{∗}(B_{i}) = ∑_{S⊂{x_{1},... ,x_{r}}} B_{i−∑_{x∈S} n_{x}+|S|}^{S} − ∑_{n_{x}=i^{x∈{x_{1},... ,xr}}} ψ_{x}.

To read this formula correctly, when i = n/2 the first sum is taken only over the subsets S ⊂ {x_{1}, …, x_{r}} containing x_{1}, that is, we do not count both S and S^{c}. Moreover we make the convention that B_{j}^{S} = 0 whenever j < |S| or j ≥ m − 1. Now replacing each ψ_{x} by its average formula from Lemma 1 we get the desired formula.

We have already seen that \overline{NE}_{1}(\overline{M}_{g,n}) is spanned by F -curves whenever g+n ≤ 7. We now present an entirely combinatorial computation of \overline{NE}_{1}(\overline{M}_{6,2}) which can serve as a model for other cases when g + n is rather small.

Proposition 8. Every F -divisor on \overline{M}_{0,8}/S_{6} is linearly equivalent to an effective combination of boundary divisors. It follows that \overline{NE}_{1}(\overline{M}_{6,2}) is generated by F - curves.

Proof. Let us denote by X := \overline{M}_{0,8}/S_{6} and by x, y ∈ \{1, …, 8\} the marked points on which S_{6} fails to act. We identify divisors on X with S_{6}-invariant divisors on \overline{M}_{0,8}. A basis for Pic(X) is given by the following collection of divisor classes:

δ_{x1} := ∑_{a≠x,y} δ_{xa}, δ_{y1} := ∑_{a≠x,y} δ_{ya}, δ_{x2} := ∑_{a,b∈{x,y}^{c}} δ_{xab}, δ_{y2} := ∑_{a,b∈{x,y}^{c}} δ_{yab},

δ_{2} := ∑_{a,b∈{x,y}^{c}} δ_{ab}, δ_{3} := ∑_{a,b,c∈{x,y}^{c}} δ_{abc}, δ_{xy1} := ∑_{a≠x,y} δ_{axy},

δ_{xy2} = δ_{4} := ∑_{a,b∈{x,y}^{c}} δ_{xyab} and δ_{x3} = δ_{y3} := ∑_{a,b,c∈{x,y}^{c}} δ_{xabc}.

There is a 10-th S_{6}-invariant divisor class on \overline{M}_{0,6}, namely δ_{xy}, which can be expressed in this basis using our average formula from Proposition 1:

δ_{xy} = \frac{1}{6}(δ_{x1} + δ_{y1}) + \frac{4}{15}(δ_{x2} + δ_{y2}) + \frac{3}{20}δ_{x3} − \frac{1}{5}δ_{3} − \frac{2}{3}δ_{xy1} − \frac{2}{5}δ_{xy2} − \frac{1}{15}δ_{2}. (7)

We now start with an arbitrary F -divisor on X:

D ≡ b_{x1}δ_{x1} + b_{y1}δ_{y1} + b_{x2}δ_{x2} + b_{y2}δ_{y2} + b_{x3}δ_{x3} + b_{xy1}δ_{xy1} + b_{xy2}δ_{xy2} + b_{2}δ_{2} + b_{3}δ_{3}. (8)

The coefficients of D are subject to 28 F -inequalities coming from all equivalence classes of partitions of {1, …, 8} in four subsets modulo the S_{6} action. By (3_{x}, 2_{y}, 2, 1) for instance we shall denote a partition of type (3 : 2 : 2 : 1) such that x is contained in the subset with 3 elements and y is part of one the subsets with 2 elements. The proof that D is linearly equivalent to an effective boundary consist of two parts and is similar to the proof of Theorem 2:

(i) If b_{2} ≥ 0 then then all coefficients in (8) are nonnegative. (ii) If b_{2} < 0 then we substitute δ_{2} using (7) and we call the resulting divisor class D_{2} ≡ D. We then take the linear combination D′ := 5D + D_{2} ≡ 6D and we show that the resulting expression,

D′ ≡ (6b_{x1} + \frac{5}{2}b_{2})δ_{x1} + (6b_{y1} + \frac{5}{2}b_{2})δ_{y1} + (6b_{x2} + 4b_{2})δ_{x2} + (6b_{y2} + δ_{2})δ_{y2} + (9)

+(6b_{x3} + \frac{9}{4}b_{2})δ_{x3} + (6b_{xy1} − 10b_{2})δ_{xy1} + (6b_{xy2} − 6b_{2})δ_{xy2} + (6b_{3} − 3b_{2})δ_{3} − 15b_{2}δ_{xy},

is effective. We present step (ii), step (i) being similar, only simpler.

We thus assume that b_{2} < 0 and we show that all coefficients in (9) are nonnegative. We start with the δ_{xy2} coefficient whose nonnegativity follows from the F -inequality corresponding to the partition (2_{xy}, 2, 2, 2). The fact that the δ_{3} coefficient is ≥ 0 comes using (2_{xy}, 2, 2, 2) and (4_{xy}, 2, 1, 1).

The other inequalities are slightly more involved. We first prove that b_{x1} ≥ 0 by combining (5_{y}, 1_{x}, 1, 1), (4_{y}, 2, 1_{x}, 1) and (3, 3, 1_{x}, 1_{y}). By symmetry we also obtain that b_{y1} ≥ 0.

Next, by adding together (4_{x}, 2_{y}, 1, 1), (4_{y}, 2_{x}, 1, 1), (3, 3, 1_{x}, 1_{y}), (3_{x}, 2_{y}, 2, 1) and (3_{y}, 2_{x}, 2, 1), we find that

4(b_{x2} + b_{y2}) + 2b_{2} ≥ 3(b_{x1} + b_{y1}) ≥ 0,

and since we have assumed b_{2} < 0 we get that b_{x2} + b_{y2} ≥ 0. Now (3_{x}, 3_{y}, 1, 1) gives that 2b_{x3} + b_{2} ≥ b_{x2} + b_{y2} ≥ 0, thus we also have that b_{x3} ≥ 0. We can now prove that the remaining coefficients in (9) are nonnegative as well.

We start with the δ_{x3} coefficient, which is nonnegative because 2b_{x3} + b_{2} ≥ 0 and b_{x3} ≥ 0. To deal with the δ_{x2} coefficient we combine (4_{y}, 2_{x}, 1, 1), (5_{y}, 1_{x}, 1, 1) with b_{x3} ≥ 0. By symmetry, the δ_{y2} coefficient is also ≥ 0. For the δ_{x1} coefficient we use that δ_{x1} ≥ 0 together with (5_{y}, 1_{x}, 1, 1). Again, by symmetry, the δ_{y1} coefficient is also ≥ 0.

We are left with the δ_{xy1} coefficient whose nonnegativity follows from b_{2} < 0 together with (4, 2_{xy}, 1, 1) and (2_{xy}, 2, 2, 2). Note that we only used 10 of the 28 F -inequalities.

We use Proposition 8 to compute the Mori cones of \overline{M}_{g,1} when g ≤ 8:

Proposition 9. The cone \overline{NE}_{1}(\overline{M}_{g,1}) is generated by F -curves for all g ≤ 8.

Proof. Since the case g ≤ 6 is settled by Proposition 5 we only need to deal with \overline{M}_{7,1} and \overline{M}_{8,1}. We only present the g = 8 case, g = 7 being similar. According to Proposition 6, it suffices to show that for any F -divisor D on \overline{M}_{0,9}/S_{8} we have that (i) D is linearly equivalent to an effective sum of boundaries, and (ii) for a boundary restriction ν : \overline{M}_{0,8} → \overline{M}_{0,9}, ν^{∗}(D) is linearly equivalent to an effective sum of boundaries.

To prove (i) we denote by Y := \overline{M}_{0,n}/S_{n−1} and by x the marked point on which S_{n−1} does not act. A basis for Pic(Y) is given by the classes δ_{i}^{{x},1} for i = 1, …, n−3. In this case the S_{n−1}-invariant boundary classes on \overline{M}_{0,n} are independent which considerably reduces the combinatorial complexity of the problem. We write (uniquely) the class of any F -divisor D on Y in this basis and the positivity of the coefficients follows in a straightforward way from the F -inequalities. We omit the details.

For (ii) it is enough to nottice that for any boundary restriction ν : \overline{M}_{0,8} → \overline{M}_{0,9} the pullback ν^{∗}(D) is an F -divisor on \overline{M}_{0,8}/S_{6} so by Proposition 8 it is equivalent to an effective sum of boundary classes.

We are now going to prove that \overline{NE}_{1}(M̃_{0,n}) is spanned by F -curves for all n ≤ 13. We use our Theorem 2 to give a Mori theoretic sufficient condition for an extremal ray on \overline{M}_{0,n} to be generated by an F -curve. The next theorem is an improvement of [KMcK], Theorem 1.2. We recall that Δ denotes the total boundary in \overline{M}_{0,n}.

Theorem 4. Let R be an extremal ray in \overline{NE}_{1}(\overline{M}_{0,n}). If there exists a nonempty effective ℚ-divisor G on \overline{M}_{0,n} such that Δ−G is also effective and (K_{\overline{M}_{0,n}}+G)·R ≤ 0, then R is contractible and it is spanned by an F -curve.

Proof. We follow the same lines as in [KMcK]. Let us write G ≡ ∑_{S} a_{S}Δ_{S}, with 0 ≤ a_{S} ≤ 1. We claim that the ray R descends to some boundary divisor. Suppose this is not the case, hence R · Δ_{S} ≥ 0 for all S and R · K_{\overline{M}_{0,n}} ≤ 0. Since there exists an ample divisor on \overline{M}_{0,n} having the same support as Δ, namely the tautological divisor κ_{1}, we find that R is generated by a contractible curve C. We denote by f : \overline{M}_{0,n} → Y the contraction. The curve C does not come from the boundary hence f_{|Δ} is finite and we can apply [KMcK], Proposition 2.5 to conclude that the exceptional locus Exc(f) is a curve. On the other hand, using the deformation theoretic bound for the dimension of the Hilbert scheme (cf. [K], Theorem 1.14)

dim_{[C]}Hilb(\overline{M}_{0,n}) ≥ −K_{\overline{M}_{0,n}} · C + n − 6 ≥ 1 (for n ≥ 7),

we conclude that C deforms inside \overline{M}_{0,n} which contradicts that Exc(f) is a curve. This argument breaks down for n ≤ 6 but in that case we can invoke directly Theorem 2 and finish the proof. Thus in any case we may assume that R is contained in some boundary divisor Δ_{T} and since \overline{NE}_{1}(Δ_{T}) = \overline{NE}_{1}(\overline{M}_{0,|T |+1})×\overline{NE}_{1}(\overline{M}_{0,|T^{c}|+1}) we may as well assume that say, R ⊆ \overline{NE}_{1}(\overline{M}_{0,|T |+1}).

If m := |T | + 1, we denote by ν : \overline{M}_{0,m} → \overline{M}_{0,n} the corresponding boundary restriction and by x the point of attachment of the fixed (|T^{c}| + 1)-pointed rational curve. We replace G by the effective divisor G′ := G + (1 − a_{T})Δ_{T}. The boundary Δ_{T} has anti-nef normal bundle hence (K_{\overline{M}_{0,n}} + G′) · R ≤ 0.

By adjunction, ν^{∗}(K_{\overline{M}_{0,n}}) = K_{\overline{M}_{0,m}} +ψ_{x}, while according to Proposition 7 we have that ν^{∗}(G′) = G̃−ψ_{x}, where G̃ is an effective divisor such that Δ_{\overline{M}_{0,m}} −G̃ is effective too. Thus (K_{\overline{M}_{0,m}} + G̃) · R ≤ 0, that is, we have exactly the initial situation on a lower dimensional moduli space and the conclusion follows inductively.

Now we show that for n ≤ 13 every extremal ray on M̃_{0,n} satisfies the conditions from Theorem 4. We start with an extremal ray R ⊆ \overline{NE}_{1}(M̃_{0,n}) and denote by E a supporting nef divisor of R. Proposition 5 gives that E is big, that is, E ∈ int(\overline{NE}^{1}(\overline{M}_{0,n})). Since K_{\overline{M}_{0,n}} = ∑_{j=2}^{⌊ \frac{n}{2} ⌋}(j(n − j)/(n − 1) − 2)B_{j}, clearly −K_{\overline{M}_{0,n}} is not effective for n ≥ 7. Following Keel and McKernan we intersect the line segment in NS(\overline{M}_{0,n}) joining −K_{\overline{M}_{0,n}} and E with the boundary of \overline{NE}^{1}(M̃_{0,n}) to get a symmetric boundary class Δ_{E} such that λE ≡ K_{\overline{M}_{0,n}} + Δ_{E} for some λ > 0. We can

write Δ_{E} ≡ ∑_{i=2}^{⌊ \frac{n}{2} ⌋} r_{i}B_{i}, where r_{i} ≥ 0 (cf. Proposition 5). That Δ_{E} is on an extremal face of the cone means r_{i} = 0 for at least one i with 2 ≤ i ≤ ⌊n/2⌋. If we can prove that r_{i} ≤ 1 for all i, then Theorem 4 gives that R is generated by an F -curve.

To achieve this we write out all F -inequalities for the nef divisor K_{\overline{M}_{0,n}} + Δ_{E}: We define the function f(a, b, c, d) to be 2 minus the number of variables equal to 1. For any partition (a, b, c, d) of n into positive integers we consider the associated F -curve given by a boundary restriction ν : \overline{M}_{0,4} → \overline{M}_{0,n}. Then using (6),

(K_{\overline{M}_{0,n}} + Δ_{E}) · ν(\overline{M}_{0,4}) = f(a, b, c, d) + r_{a+b} + r_{a+c} + r_{a+d} − r_{a} − r_{b} − r_{c} − r_{d} ≥ 0.

Theorem 5. For n ≤ 13 any nontrivial nef divisor on M̃_{0,n} is of the form K_{\overline{M}_{0,n}} + Δ_{E}, with 0 ≤ Δ_{E} ≤ Δ. It follows that \overline{NE}_{1}(\overline{M}_{g}) is generated by F -curves for all g ≤ 13.

Proof. We start with the nef divisor K_{\overline{M}_{0,n}} + Δ_{E}, where Δ_{E} ≡ ∑_{j=2}^{⌊n/2⌋} r_{j}B_{j}, where r_{j} ≥ 0 for all j and there is 2 ≤ i ≤ ⌊n/2⌋ such that r_{i} = 0. By using all F - inequalities the coefficients r_{j} are subject to, we conclude that r_{j} ≤ 1. We carry this out only for n = 13, the case n = 12 being entirely similar. We list all F -inequalities for n = 13:

(1) 3r_{2} ≥ r_{3} + 1, (2) 2r_{3} ≥ r_{4}, (3) r_{2} + 2r_{4} ≥ r_{3} + r_{5}, (4) r_{2} + 2r_{5} ≥ r_{4} + r_{6},

(5) r_{2} + r_{6} ≥ r_{5}, (6) 1 + 2r_{3} + r_{4} ≥ 2r_{2} + r_{5}, (7) 1 + r_{4} + r_{5} ≥ r_{2} + r_{6},

(8) 1 + r_{3} + r_{5} ≥ r_{2} + r_{4}, (9) 1 + r_{3} + 2r_{6} ≥ r_{2} + 2r_{5}, (10) 1 + 2r_{4} ≥ 2r_{3},

(11) 1 + r_{6} ≥ r_{3}, (12) 1 + 3r_{5} ≥ 3r_{4}, (13) 2 + 3r_{4} ≥ 3r_{2} + r_{6},

(14) 2+r_{4} +2r_{5} ≥ 2r_{2} +r_{3} +r_{6}, (15) 2+2r_{6} ≥ 2r_{2} +r_{5}, (16) 2+r_{5} +r_{6} ≥ r_{2} +2r_{3},

(17) 2 + r_{5} + 2r_{6} ≥ r_{2} + r_{3} + 2r_{4}, and finally (18) 2 + 3r_{6} ≥ 3r_{3} + r_{4}.

From (1) we see that r_{2} > 0. We have four cases:

(i) r_{3} = 0. Then from (2) we have r_{4} = 0 while from (6) we get r_{5} < 1 and r_{2} ≤ 1/2. Assume now that r_{6} ≥ 1. Then (3) and (7) combined give r_{5} = r_{2} and r_{6} = 1. From (13) we get that r_{2} ≤ 1/3 while (4) gives that r_{2} ≥ 1/3, hence r_{2} = r_{5} = 1/3. Thus either r_{j} < 1 for all j or else Δ_{E} ≡ \frac{1}{3}(B_{2} + B_{5}) + B_{6}. (ii) r_{4} = 0. Use (10), (13) and (3) to get that r_{3} ≤ 1/2, r_{2} ≤ 2/3 and r_{5} ≤ 2/3. We assume again that r_{6} ≥ 1. Then (3) and (7) are compatible only when r_{2} = r_{5} and r_{3} = 0, that is, we are back to case (i). (iii) r_{5} = 0. Inequality (12) yields r_{4} ≤ 1/3 while from (13) we get that r_{2} ≤ 1. Moreover if r_{2} = 1 then r_{6} = 0, r_{4} = 1/3 and r_{3} = 1/3, so in this way we get our second exceptional case, Δ_{E} ≡ B_{2} + \frac{1}{3}(B_{3} + B_{4}). On the other hand if r_{2} < 1 then from (4) we have that r_{6} < 1 and finally from (10) we obtain that r_{3} ≤ 5/6.

(iv) r_{6} = 0. From (18) we have that r_{3} ≤ 2/3 while (5) and (15) give that r_{5} ≤ 2/3 and r_{2} ≤ 1. Moreover r_{2} = 1 implies r_{5} = 0 so we are back to case (iii). Then we use (12) which gives r_{4} ≤ 1. If r_{4} = 1 from (18) we have r_{3} ≤ 1/3 while from (2) r_{3} ≥ 1/2, a contradiction, so this last case does not occur.

Remark. For n ≤ 11 every nontrivial F -class on M̃_{0,n} is of the form K_{\overline{M}_{0,n}} + Δ_{E}, where Δ_{E} is a pure boundary, that is, Δ_{E} = ∑_{i} r_{i}B_{i} where 0 ≤ r_{i} < 1 (cf. [KMcK], Corollary 5.3). The previous proof shows that on M̃_{0,13} there are exactly two F - classes not of this form:

K_{\overline{M}_{0,13}} + \frac{1}{3}(B_{2} + B_{5}) + B_{6} and K_{\overline{M}_{0,13}} + B_{2} + \frac{1}{3}(B_{3} + B_{4}),

On M̃_{0,12} there is just one such class, namely K_{\overline{M}_{0,12}} + (B_{2} + B_{5})/3 + B_{6}. For n ≥ 14 it is no longer true that any nontrivial F - divisor on M̃_{0,n} is numerically equivalent to K_{\overline{M}_{0,n}} + Δ_{E}, where Δ_{E} = ∑_{i} r_{i}B_{i} with 0 ≤ r_{i} ≤ 1. When n = 14 for instance, the F -class

K_{\overline{M}_{0,14}} + \frac{1}{3}(B_{2} + B_{5}) + B_{6} + r_{7}B_{7}

where r_{7} ∈ [1, 4/3], is not of this form.

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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN, ANN ARBOR, MI 48109-1109

    E-mail: gfarkas@umich.edu

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN, ANN ARBOR, MI 48109-1109

    E-mail: agibney@umich.edu