EFFECTIVE DIVISORS ON \overline{M}_{g} AND A COUNTEREXAMPLE TO THE SLOPE CONJECTURE

GAVRIL FARKAS AND MIHNEA POPA

1. Introduction

The purpose of this note is to prove two statements on the slopes of effective divisors on the moduli space of stable curves \overline{M}_{g}: first that the Harris-Morrison Slope Conjecture fails to hold on \overline{M}_{10} and second, that in order to compute the slope of \overline{M}_{g} for g ≤ 23, one only has to look at the coefficients of the classes λ and δ_{0} in the expansion of the relevant divisors. The proofs are based on a general result providing inequalities between the first few coefficients of effective divisors on \overline{M}_{g}. We give the technical statements in what follows.

On \overline{M}_{g} we denote by λ the class of the Hodge line bundle, by δ_{0}, …, δ_{[g/2]} the boundary divisor classes corresponding to singular stable curves and by δ := δ_{0} + ⋯ + δ_{[g/2]} the total boundary. If 𝔼 ⊂ Pic(\overline{M}_{g}) ⊗ ℝ is the effective cone, then following [HMo] we define the slope function s : 𝔼 → ℝ ∪ {∞} by the formula

s(D) := inf \{\frac{a}{b} : a, b > 0 such that aλ − bδ − D ≡ ∑_{i=0}^{[g/2]} c_{i}δ_{i}, where c_{i} ≥ 0\}.

From the definition it follows that s(D) = ∞ unless D ≡ aλ − ∑_{i=0}^{[g/2]} b_{i}δ_{i} with a, b_{i} ≥ 0 for all i (and it is well-known that s(D) < ∞ for any D which is the closure of an effective divisor on M_{g}). In the second case one has that s(D) = a/min_{i=0}^{[g/2]}b_{i}. We denote by s_{g} the slope of the moduli space \overline{M}_{g}, defined as s_{g} := inf \{s(D) : D ∈ 𝔼\}. The Slope Conjecture of Harris and Morrison predicts that s_{g} ≥ 6 + 12/(g + 1) (cf.

[HMo] Conjecture 0.1). This is known to hold for g ≤ 12, g ≠ 10 (cf. [HMo] and [Ta]).

Following [CU], we consider the divisor K on M_{10} consisting of smooth curves lying on a K3 surface, and we denote by \overline{K} its closure in \overline{M}_{10}. For any g ≥ 20, we look at the locus in \overline{M}_{g} of curves obtained by attaching a pointed curve of genus g − 10 to a curve in \overline{K} with a marked point. This gives a divisor in Δ_{10} ⊂ \overline{M}_{g}, which we denote by Z.

Research of GF partially supported by the NSF Grant DMS-0140520.

Research of MP partially supported by the NSF Grant DMS-0200150.

We thank Joe Harris and Rob Lazarsfeld for interesting discussions on this subject.

The key point in what follows is that, based on the study of curves lying on K3 surfaces, one can establish inequalities involving a number of coefficients of any effective divisor coming from M_{g} in the expansion in terms of the generating classes.

Theorem 1.1. Let D ≡ aλ − ∑_{i=0}^{[g/2]} b_{i}δ_{i} be the closure in \overline{M}_{g} of an effective divisor on M_{g}.

(a) For 2 ≤ i ≤ 9 and i = 11 we have b_{i} ≥ (6i + 18)b_{0} − (i + 1)a. The same formula holds for i = 10 if D does not contain the divisor Z ⊂ Δ_{10}.

(b) (g ≥ 20) If D contains Z, then either b_{10} ≥ 78b_{0}−11a as above, or b_{10} ≥ (71.3866...)· b_{0} − (10.1980...) · a.

(c) We always have that b_{1} ≥ 12b_{0} − a.

Part (a) (and (c)) of this theorem essentially only make more concrete results – and the technique of intersecting with special Lefschetz pencils– already existing in the literature mentioned above. Part (b) however is more involved: it requires pull-backs to \overline{M}_{10,1} and the intrinsic use of our partial knowledge about the divisor \overline{K}, plus some facts about the Weierstrass divisor on \overline{M}_{g,1}. Here we claim more originality.

Corollary 1.2. If a/b_{0} ≤ 71/10, then b_{i} ≥ b_{0} for all 1 ≤ i ≤ 9. The same conclusion holds for i = 10 if a/b_{0} ≤ 6.906 …, and for i = 11 if a/b_{0} ≤ 83/12.

Based on this we obtain that the divisor \overline{K} ⊂ \overline{M}_{10} provides a counterexample to the Slope Conjecture. Its class can be written as

\overline{K} ≡ aλ − b_{0}δ_{0} − … − b_{5}δ_{5},

and by [CU] Proposition 3.5, we have a = 7 and b_{0} = 1. In view of Corollary 1.2, this information is sufficent to show that the slope of \overline{K} is smaller than the one expected based on the Slope Conjecture.

Corollary 1.3. The slope of \overline{K} is equal to a/b_{0} = 7, so strictly smaller than the bound 78/11 predicted by the Slope Conjecture. In particular s_{10} = 7 (since by [Ta] s_{10} ≥ 7).

Theorem 1.1 also allows us to formulate (at least up to genus 23, and conjecturally beyond that) the following principle: the slope s_{g} of \overline{M}_{g} is computed by the quotient a/b_{0} of the relevant divisors. We have more generally:

Theorem 1.4. For any g ≤ 23, there exists ϵ_{g} > 0 such that for any effective divisor D on \overline{M}_{g} with s_{g} ≤ s(D) ≤ s_{g} + ϵ_{g} we have s(D) = a/b_{0}, i.e. b_{0} ≤ b_{i} for all i ≥ 1.

Conjecture 1.5. The statement of the theorem holds in arbitrary genus.

Theorem 1.1 is proved in §2. Theorem 1.4 is proved in §3, where we also remark that the methods of the present paper give a very quick proof of the fact that the Kodaira dimension of the universal curve \overline{M}_{g,1} is −∞ for g ≤ 15, g ≠ 13, 14.

EFFECTIVE DIVISORS ON \overline{M}_{g} 3

2. Inequalities between coefficients of divisors

Let F_{g} be the moduli space of canonically polarized K3 surfaces (S, H) of genus g. We consider the P^{g}-bundle P_{g} = {(S, C) : C ∈ |H|} over F_{g} which comes with a natural rational map ϕ_{g} : P_{g} − − > M_{g}. By Mukai’s results [Mu1] and [MM], this map is dominant if and only if 2 ≤ g ≤ 9 or g = 11. In this range M_{g} can be covered by curves corresponding to Lefschetz pencils of curves on K3 surfaces (cf. [Ta]). This is not true any more when g = 10: in this case Im(ϕ_{10}) is a divisor K in M_{10} (cf. [CU] Proposition 2.2).

Note that any such Lefschetz pencil, considered as a family of curves over P^{1}, has at least one section, since its base locus is nonempty.

Given 2 ≤ i ≤ 11, consider as above a Lefschetz pencil of curves of genus i lying on a general K3 surface of degree 2i−2 in P^{i}. This gives rise to a curve B in the moduli space \overline{M}_{i}.

Lemma 2.1. We have the formulas B · λ = i + 1, B · δ_{0} = 6i + 18 and B · δ_{j} = 0 for j ≠ 0.

Proof. The first two numbers are computed e.g. in [CU] Proposition 3.1, based on the formulas in [GH] pp. 508–509. The last assertion is obvious since there are no reducible curves in a Lefschetz pencil. □

For each g ≥ i+1, starting with the pencil B in \overline{M}_{i} we can construct a new pencil B_{i} in \overline{M}_{g} in the following way: we fix a general pointed curve (C, p) genus g − i. We then glue the curves in the pencil B with C at p, along one of the sections corresponding to the base points of the pencil. We have that all such B_{i} fill up Δ_{i} ⊂ \overline{M}_{g} for i ≠ 10, and the divisor Z ⊂ Δ_{10} when i = 10.

Lemma 2.2. We have B_{i} · λ = i + 1, B_{i} · δ_{0} = 6i + 18, B_{i} · δ_{i} = −1 and B_{i} · δ_{j} = 0 for j ≠ 0, i.

Proof. This follows immediately from Lemma 2.1 and from general principles, as explained in [CR] pp.271. □

Proof. (of Theorem 1.1 (a), (c)) (a) Let us fix 2 ≤ i ≤ 11, i ≠ 10. Since D is the closure of a divisor coming from M_{g}, it cannot contain the whole boundary Δ_{i}. Thus we must have a pencil B_{i} as above such that B_{i} · D ≥ 0. The same thing holds true for i = 10 if we know that Z is not contained in D. But by Lemma 2.2 this is precisely the statement of this part.

(c) We follow the same procedure, but this time we produce a pencil B_{1} in Δ_{1} ⊂ \overline{M}_{g} by gluing a fixed pointed curve (C, p) of genus g − 1 to a generic pencil of plane cubics along one of its 9 sections. We have the well-known relations:

B_{1} · λ = 1, B_{1} · δ_{0} = 12, B_{1} · δ_{1} = −1 and B_{1} · δ_{j} = 0 for j ≠ 0, 1.

The conclusion follows similarly, since we can find a B_{1} such that B_{1} · D ≥ 0. □

The study of the coefficient b_{10} is more involved, since in M_{10} the Lefschetz pencils of curves on K3 surfaces only fill up a divisor. We need some preliminaries on divisors on the universal curve \overline{M}_{g,1}. Let π : \overline{M}_{g,1} → \overline{M}_{g} be the forgetful morphism. The generators of Pic(\overline{M}_{g,1}) ⊗ ℚ are the tautological class ψ = c_{1}(ω_{π}), the boundary δ_{0}, the Hodge class λ, and for 1 ≤ i ≤ g − 1 the class δ_{i} corresponding to the locus of pointed curves consisting of two components of genus i and g − i respectively with the marked point being on the genus i component.

Lemma 2.3. (cf. [AC] §1) One has the following relations:

π_{∗}(λ^{2}) = π_{∗}(λ · δ_{i}) = π_{∗}(δ_{0} · δ_{i}) = 0 for all i = 0, …, g − 1, π_{∗}(ψ^{2}) = 12λ − δ,
π_{∗}(λ · ψ) = (2g − 2)λ, π_{∗}(ψ · δ_{0}) = (2g − 2)δ_{0}, π_{∗}(ψ · δ_{i}) = (2i − 1)δ_{i} for i ≥ 1,
π_{∗}(δ_{i}^{2}) = −δ_{i} for 1 ≤ i ≤ g − 1, π_{∗}(δ_{i} · δ_{g−i}) = δ_{i}, for 1 ≤ i < g/2, and
π_{∗}(δ_{i} · δ_{j}) = 0 for all i, j ≥ 0 with i ≠ j, g − j.

We consider the Weierstrass divisor in M_{g,1}

W := {[C, p] ∈ M_{g,1} : p ∈ C is a Weierstrass point},

and denote by \overline{W} its closure in \overline{M}_{g,1}. Its class has been computed by Cukierman [Ck]:

\overline{W} ≡ −λ + \frac{g(g + 1)}{2}ψ − ∑_{i=1}^{g−1} \binom{g − i + 1}{2}δ_{i}.

Proposition 2.4. If π : \overline{M}_{g,1} → \overline{M}_{g} is the forgetful morphism, then π_{∗}(\overline{W}^{2}) is an effective divisor class on \overline{M}_{g}.

Proof. From the previous Lemma we have that

π_{∗}(\overline{W}^{2}) ≡ aλ − ∑_{i=0}^{[g/2]} b_{i}δ_{i},

where a = g(g + 1)(3g^{2} + g + 2), b_{0} = g^{2}(g + 1)^{2}/4 while for 1 ≤ i < g/2 we have b_{i} = i(g − i)(g^{3} + 3g^{2} + g − 1). When g is even b_{g/2} = (8g^{5} + 28g^{3} + 33g^{4} + 4g^{2})/64. On the other hand we have expressions for the classes of distinguished geometric divisors on \overline{M}_{g}: when g + 1 is composite, by looking at Brill-Noether divisors one sees that the class

(g + 3)λ − \frac{g + 1}{6}δ_{0} − ∑_{i=1}^{[g/2]} i(g − i)δ_{i}

is effective (cf. [EH] Theorem 1). When g + 1 is prime one has to use the class of the Petri divisor, which gives a slightly worse estimate (cf. [EH] Theorem 2). In either case, by comparing the coefficients a, b_{i} above with those of these explicit effective classes, one obtains an effective representative for π_{∗}(\overline{W}^{2}). For instance when g + 1 is composite it is enough to check that b_{0}/a ≤ (g + 1)/(6g + 18) and that b_{i}/a ≤ i(g − i)/(g + 3) for i = 1, …, [g/2], which is immediate. □

Corollary 2.5. Let D be any effective divisor class on \overline{M}_{g,1}. Then π_{∗}(\overline{W} · D) is an effective class on \overline{M}_{g}.

Proof. Since \overline{W} is irreducible we can write D = m\overline{W} +E, where E is an effective divisor not containing \overline{W} and m ∈ ℤ_{≥0}. Then we use the previous Proposition. □

Proof. (of Theorem 1.1 (b)) Assume that b_{10} < 78b_{0} − 11a. We consider the map

j : \overline{M}_{10,1} ⟶ \overline{M}_{g}

obtained by attaching a fixed general pointed curve of genus g −10 to any curve of genus 10 with a marked point. Our assumption says that R · j^{∗}(D) < 0, where R ⊂ \overline{M}_{10,1} denotes the curve in the moduli space coming from a Lefschetz pencil of pointed curves of genus 10 on a general K3 surface. We can write j^{∗}(D) = mπ^{∗}(\overline{K}) + E, where E is an effective divisor not containing π^{∗}(\overline{K}) and m ∈ ℤ is such that

(1)  m ≥ −R · j^{∗}(D) = −11a + 78b_{0} − b_{10} > 0.

Note that we have the standard formulas j^{∗}(λ) = λ, j^{∗}(δ_{0}) = δ_{0} and j^{∗}(δ_{10}) = −ψ (cf. [AC]), while \overline{K} ≡ 7λ − δ_{0} − ⋯, hence

E ≡ (a − 7m)λ + b_{10}ψ + (m − b_{0})δ_{0} + (other boundaries)

is an effective class on \overline{M}_{10,1}. By applying Corollary 2.5 it follows that π_{∗}(\overline{W} · E) is an effective class on \overline{M}_{10}. An easy calculation using Lemma 2.3 shows that

π_{∗}(\overline{W} · E) ≡ (642b_{10} + 990(a − 7m))λ − 55(b_{10} + 18(b_{0} − m))δ_{0} − · · · .

We now use the fact that for every effective divisor on \overline{M}_{g} the coefficient a of λ is nonnegative ^{1}. From the previous formula we get an inequality which combined with (1) yields, after a simple computation

b_{10} ≥ (71.3866...) · b_{0} − (10.1980...) · a.
□

Question 2.6. For reasons of uniformity, it is natural to ask the following: does the second situation in Theorem 1.1(b) actually occur, or do we always have even for b_{10} the same inequality as in part (a)?

We conclude with some examples where these inequalities can be checked directly and are sometimes sharp.

1For the reader’s convenience we recall that this follows immediately from the fact that B · λ > 0 for any curve B ⊂ \overline{M}_{g} such that B ∩ M_{g} ≠ ∅, while there is always a complete curve in M_{g} passing through a general point.

Example 2.7. When g+1 is composite, if r, d > 0 are such that g+1 = (r+1)(g−d+r), then the locus of curves of genus g carrying a 𝔤_{d}^{r} is a divisor with class (cf. [EH], Theorem 1):

\overline{M}_{g,r}^{d} ≡ c((g + 3)λ − \frac{g + 1}{6}δ_{0} − ∑_{i=1}^{[g/2]} i(g − i)δ_{i}),

where c is a positive constant depending on g, r and d. A simple calculation shows that the inequalities in Theorem 1.1 are satisfied. Moreover, they are sharp for i = 1 and i = 2, for any genus g.

Example 2.8. A similar behavior is exhibited by the divisor of curves on K3 surfaces \overline{K} ⊂ \overline{M}_{10}, where we have b_{1} = 5 and b_{2} = 9 which again gives equality in the first two inequalities in Theorem 1.1. Note that this follows by the method of [EH] §2 if we show that the pull-back of \overline{K} to \overline{M}_{2,1} is supported on the Weierstrass divisor. We will obtain this in the forthcoming paper [FP], based on results of Voisin [V], and as a special case of a more general study of degenerations of spaces of sections of rank two vector bundles on curves. The same study will show a striking difference between the geometry of \overline{K} and that of the Brill-Noether divisors, namely that the image of the natural map from \overline{M}_{0,g} to \overline{M}_{g} is contained in the K3-locus for any g. Thus one cannot use the method of [EH] §3 in order to determine more coefficents of \overline{K}.

3. Slopes of divisors and further remarks

The inequalities established in Theorem 1.1 allow us to show that, at least up to genus 23, if the slope of an effective divisor is sufficiently small, then it is computed by the ratio a/b_{0}.

Proof. (of Theorem 1.4) When g is such that g + 1 is composite, we have that s_{g} ≤ 6 + 12/(g + 1) (this being the slope of any Brill-Noether divisor). When g is even, one has the estimate s_{g} ≤ \frac{2(3g^{2}+13g+2)}{g(g+2)} (this being the slope of the Petri divisor, cf. [EH] Theorem 2). It follows that for any g ≤ 23 there exists a positive number ϵ_{g} such that

s_{g} + ϵ_{g} ≤ 6 + \frac{11}{i + 1} for all i ≤ [g/2](≤ 11).

Assume first that 2 ≤ i ≤ 9 or i = 11. Then by Theorem 1.1(a) we know that b_{i} ≥ (6i + 18)b_{0} − (i + 1)a, and so certainly b_{i} ≥ b_{0} if s(D) ≤ 6 + \frac{11}{i+1}. For i = 10 we apply 1.1(b): if the inequality b_{10} ≥ 78b_{0} − 11a holds, then the argument is identical. If not, we have the inequality b_{10} ≥ (71.3866...) · b_{0} − (10.1980...) · a. Thus b_{10} ≥ b_{0} as soon as the inequality a/b_{0} ≤ 6.9 is satisfied. But for g ≥ 20 the inequality s_{g} < 6.9 holds, based on the same estimates as above.

For i = 1, the condition is even weaker because of the formula b_{1} ≥ 12b_{0} − a in 1.1(c). Thus the slope of D is computed by a/b_{0}. □

Remark 3.1. Let us consider again the curve B ⊂ \overline{M}_{g} corresponding to a Lefschetz pencil of curves of genus g on a general K3 surface (cf. Lemma 2.1). Since B ·δ_{0}/B ·λ = 6 + 12/(g + 1) which is the conjectured value of s_{g}, it follows that the nefness ^{2} of B would be a sufficient condition for the Slope Conjecture to hold in genus g. Moreover, Theorem 1.4 and Corollary 1.2 imply that for g ≤ 23 the Slope Conjecture in genus g is equivalent to B being a nef curve. The conjecture fails for g = 10 because B · \overline{K} = −1. Remark 3.2. An amusing consequence of Proposition 2.5 is that the Kodaira dimension of the universal curve M_{g,1} is −∞ for all g ≤ 15, with g ≠ 13, 14 ( of course this can be proved directly when g ≤ 11). Indeed, if we assume that the canonical class K_{\overline{M}_{g,1}} ≡ 13λ + ψ − 3(δ_{1} + δ_{g−1}) − 2 ∑_{i=2}^{g−2} δ_{i} is effective on \overline{M}_{g,1}, then by Proposition 2.5

the class D := π_{∗}(K_{\overline{M}_{g,1}}·\overline{W}) is effective on \overline{M}_{g}. It turns out that s(D) = \frac{2(13g^{3}+6g^{2}−9g+2)}{g(g+1)(4g+3)}, and from the definition of the slope of M_{g} we have that s(D) ≥ s_{g}. But this contradicts the estimates on s_{g} from [Ta] and [CR].

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[MM] S. Mori and S. Mukai, The uniruledness of the moduli space of curves of genus 11, Lecture Notes in Mathematics 1016 (1983), 334-353.

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[V] C. Voisin, Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, Acta Math., 168 (1992), 249-272.

Department of Mathematics, University of Michigan, 525 East University, Ann Arbor, MI, 48109-1109

      E-mail address: gfarkas@umich.edu

Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138

      E-mail address: mpopa@math.harvard.edu

2Recall that, slightly abusively, a curve B on a projective variety X is called nef if B · D ≥ 0 for every efective divisor D on X.