EFFECTIVE DIVISORS ON \overline{M}_{g}, CURVES ON K3 SURFACES, AND THE SLOPE CONJECTURE

GAVRIL FARKAS AND MIHNEA POPA

1. Introduction

In this paper we use the geometry of curves lying on K3 surfaces in order to obtain a number of results about effective divisors on the moduli space of stable curves \overline{M}_{g}. We begin by showing two statements on the slopes of such divisors: 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 standard expansion in terms of the generators of the Picard group. The proofs are based on a general result providing inequalities between the first few coefficients of effective divisors on \overline{M}_{g}. We then describe in detail the divisor \overline{K} on M_{10} consisting of smooth sections of K3 surfaces, and its compactification \overline{K} in the moduli space \overline{M}_{10} (which is the counterexample to the conjecture mentioned above). As far as we know this is the first intersection theoretic analysis of a geometric subvariety on \overline{M}_{g} which is not of classical Brill-Noether-Petri type, that is, a locus of curves carrying an exceptional linear series 𝔤_{d}^{r}. Along the way, various other results on the Kodaira-Iitaka dimension of distinguished linear series on certain moduli spaces of (pointed) stable curves are obtained. We give the technical statements in what follows. ^{1}

On \overline{M}_{g} we denote by λ the class of the Hodge line bundle, by δ_{0}, …, δ_{[g/2]} the boundary 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 cone of effective divisors, then following [HM] 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}, see e.g. [HM]). In this 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. [HM], Conjecture 0.1). This is known to hold for g ≤ 12, g ≠ 10 (cf. [HM] and [Ta]). Apart from the evidence coming from low genus, the conjecture was mainly based on the large number of calculations of classes of geometric divisors on \overline{M}_{g} (see e.g. [EH3], [Ha]). The original paper [HM] describes a number of interesting consequences of the statement, e.g. a positive answer would imply that the Kodaira dimension of M_{g} is −∞ if and only if g ≤ 22. Proposition 2.2 below provides a geometrically meaningful refomulation of the conjecture saying that if there exists an effective divisor D on \overline{M}_{g} with slope s(D) < 6 + 12/(g + 1), then D has to contain the locus \overline{K}_{g} of K3 sections of genus g.

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

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

1Apart from the study of divisors on \overline{M}_{g}, the importance of using curves sitting on a K3 surface is well-documented: Lazarsfeld observed that there is no Brill-Noether type obstruction to embed a curve in a K3 surface, thus giving the first proof of the Brill-Noether-Petri Theorem which does not use degeneration to singular curves (cf. [La]). More recently, Voisin used the geometry of K3 sections to prove the generic Green Conjecture (cf. [V2], [V3]). Very interesting geometry has been developed by Mukai (cf. e.g. [M1], [M2]), whose results will be used below.

We consider the divisor \overline{K} on M_{10} consisting of smooth curves lying on a K3 surface, and we denote by K10 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 K10 with a marked point. This gives a divisor in Δ_{10} ⊂ \overline{M}_{g}, which we denote by Z.

Based on the study of curves lying on K3 surfaces, we establish in §2 inequalities involving a number of coefficients of any effective divisor coming from M_{g} in the expansion in terms of 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} ≥ 78 · \frac{1155}{1262} · b_{0} − 11 · \frac{1170}{1262} · a

(which means approximately b_{10} ≥ (71.3866...) · b_{0} − (10.1980...) · a).

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

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} ≤ \frac{88828}{12870} = 6.9019..., and for i = 11 if a/b_{0} ≤ 83/12.

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

K10 ≡ 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 K10 is smaller than expected.

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.

Effective divisors on \overline{M}_{g} 3

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

We next carry out (in §3 – §5) a detailed study of the compactified K3 divisor \overline{K} considered above. In the course of doing this we develop techniques (and give applications) which go beyond this example and will hopefully also find other uses. We prove the following result:

Theorem 1.6. The class of the divisor \overline{K} in Pic(\overline{M}_{10}) is given by

\overline{\overline{K}} ≡ 7λ − δ_{0} − 5δ_{1} − 9δ_{2} − 12δ_{3} − 14δ_{4} − B_{5}δ_{5},

with B_{5} ≥ 6.

Once more we see that the slope of \overline{K} is equal to 7, so strictly smaller than the bound 78/11 predicted by the Slope Conjecture. The first two coefficients in this expression were computed in [CU]. Unfortunately we are unable to pin down B_{5} with the methods of this paper. It would be very surprising though if B_{5} were not 15, as it will be clear from the discussion below. However, the specific applications in which we need \overline{\overline{K}} do not use the value of this coefficient.

To compute the class of \overline{K} we show that one can think of points in \overline{K} in four different ways, i.e. \overline{K} has four different realizations as a geometric divisor on M_{10}.

Theorem 1.7. The divisor K can be described (set-theoretically) as any of the following subvarieties of M_{10}:

(1) (By definition) The locus of curves sitting on a K3 surface.

(2) The locus of curves C with a non-surjective Wahl map ψ_{K} : ∧^{2}H^{0}(K_{C}) → H^{0}(3K_{C}).

(3) The divisorial component of the locus of curves C carrying a semistable rank two vector bundle E with ∧^{2}(E) = K_{C} and h^{0}(E) ≥ 7.

(4) The divisorial component of the locus of genus 10 curves sitting on a quadric in an embedding C ⊂ P^{4} with deg(C) = 12.

We deduce this in fact by showing the equivalence of the four descriptions over the locus of Brill-Noether general curves, whose complement has codimension 2 in M_{10}. Note that the equivalence of descriptions (1) and (2) has been proved in [CU]. We obtain the expression for the class of \overline{K} as a consequence of a more general study of the degenerations of multiplication maps for sections of line bundles on curves. This is intimately related to characterizations (3) and (4) above. It is important to emphasize the role of condition (3): it shows that the divisor \overline{K} is a higher rank Brill-Noether divisor, more precisely one attached to rank 2 vector bundles with canonical determinant. This was in fact the initial motivation for our study, and as a general method it is likely to lead to further developments. By extrapolating descriptions (3) and (4) to other genera g ≥ 13 one can construct geometric divisors on \overline{M}_{g} containing the locus \overline{K}_{g} and which we expect to provide other counterexamples to the Slope Conjecture. For instance when g = 13 the closure D in \overline{M}_{13} of the locus of curves C of genus 13 sitting on a quadric in an embedding C ⊂ P^{5},is a divisor containing \overline{K}_{13} and we expect that s(D) < 6 + 12/14. We plan to return to these problems in the future.

There is a quite striking similarity between the class of \overline{\overline{K}} in Pic(\overline{M}_{10}) and the class of the Brill-Noether divisors in the next genus g = 11: on \overline{M}_{11} there are two distinguished geometric divisors, the 6-gonal locus \overline{M}_{11,6}^{1} and the divisor \overline{M}_{11,9}^{2} of curves with a 𝔤_{9}^{2}. These are distinct irreducible divisors on \overline{M}_{11} having proportional classes (cf. [EH3], Theorem 1):

α\overline{M}_{11,6}^{1} ≡ β\overline{M}_{11,9}^{2} ≡ 7λ − δ_{0} − 5δ_{1} − 9δ_{2} − 12δ_{3} − 14δ_{4} − 15δ_{5},

for precisely determined α, β ∈ ℤ_{>0}. The reason for this resemblance is that the coefficients of these divisors are (up to a constant) the same as those of any other divisor whose pullback to \overline{M}_{i,1}, for a sufficient number of i’s less than g, is a combination of generalized Brill-Noether divisors (cf. §5 for specific details). The similarity is surprising, since \overline{K} behaves geometrically very differently from all Brill-Noether divisors. For instance, while it is known that all flag curves of genus g consisting of a rational spine and g elliptic tails are outside every Brill-Noether divisor, we prove that for every g they belong to the compactification \overline{K}_{g} in \overline{M}_{g} of the K3 locus (cf. §7). In the same context, we also look in §6 at linear systems on \overline{M}_{g} having the minimal slope 6 + 12/(g + 1) predicted by the Slope Conjecture. Namely on \overline{M}_{11}, where the Slope Conjecture is known to hold, although there exist only the two Brill-Noether divisors described above we show the following (cf. Proposition 6.2 for a more precise statement):

Proposition 1.8. There exist effective divisors on \overline{M}_{11} of slope 7 and having Iitaka dimension equal to 19.

This fact seems to contradict the hypothesis formulated in [HM] (and proved to be true for low g) that the Brill-Noether divisors are essentially the only effective divisors on \overline{M}_{g} of slope 6 + 12/(g + 1).

We conclude with a number of further applications. As a consequence of Theorem 1.6, in §7 we study the birational nature of the moduli spaces \overline{M}_{10,n} of stable genus 10 curves with n marked points:

Theorem 1.9. The Kodaira dimension of \overline{M}_{10,10} is nonnegative, while \overline{M}_{10,n} is of general type for n ≥ 11. On the other hand κ(\overline{M}_{10,n}) = −∞ for n ≤ 9.

Also, we remark already in §2 (Remark 2.10) 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.

Acknowledgments. We have benefitted from discussions with Igor Dolgachev and Joe Harris. We especially thank Sean Keel for suggesting a potential connection between the Brill-Noether linear system on \overline{M}_{11} and divisors on \overline{F}_{11}, eventually leading to Proposition 6.2, and Rob Lazarsfeld for his help in the proof of Theorem 1.7.

2. Inequalities between coefficients of divisors

In this section we give a geometric interpretation of the Slope Conjecture and we establish constraints on the coefficients of λ, δ_{0}, …, δ_{11} for any effective divisor on \overline{M}_{g}.

Given g ≥ 1 we consider a Lefschetz pencil of curves of genus g lying on a general K3 surface of degree 2g − 2 in P^{g}. This gives rise to a curve B in the moduli space \overline{M}_{g}. 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. Such pencils B fill up the entire moduli space \overline{M}_{g} for g ≤ 9 or g = 11 (cf. [M1] , [M2]) and the divisor \overline{K} for g = 10. For g ≥ 13, the pencil B fills up the locus \overline{K}_{g} of K3 sections of genus g and dim(\overline{K}_{g}) = 19+g.

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

Proof. The first two numbers are computed using classical formulas from [GH], pp. 508– 509. The last assertion is obvious since there are no reducible curves in a Lefschetz pencil. □

Proposition 2.2. Let D be the closure in \overline{M}_{g} of an effective divisor on M_{g}. If s(D) < 6 + 12/(g + 1), then D contains the K3 locus \overline{K}_{g}.

Proof. We consider as above the curve B ⊂ \overline{M}_{g} corresponding to a Lefschetz pencil of curves of genus g on a general K3 surface S. From Lemma 2.1 we obtain that B · δ/B · λ = 6 + 12/(g + 1) > s(D), which implies that B · D < 0 hence B ⊂ D. By varying B and S we obtain that \overline{K}_{g} ⊂ D. □

Remark 2.3. Proposition 2.2 shows that the nefness ^{2} of B would be a sufficient condition for the Slope Conjecture to hold in genus g. Moreover, note (a bit prematurely) that Theorems 1.4 and 1.1 imply that for g ≤ 23 the Slope Conjecture in genus g is equivalent to B being a nef curve. The conjecture will fail for g = 10 precisely because B · \overline{K} = −1.

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} as follows: 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 2.4. 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 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 2.4 this is precisely the statement of this part.

(c) This is undoubtedly well known. □

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

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 \overline{M}_{g,n}. For 0 ≤ i ≤ g and S ⊂ \{1, …, n\}, the boundary divisor Δ_{i:S} corresponds to the closure of the locus of nodal curves C_{1} ∪ C_{2}, with C_{1} smooth of genus i, C_{2} smooth of genus g − i, and such that the marked points sitting on C_{1} are precisely those labelled by S. We also consider the divisor Δ_{irr} consisting of irreducible pointed curves with one node. We denote by δ_{i:S} ∈ Pic(\overline{M}_{g,n}) the class of Δ_{i:S} and by δ_{irr} that of Δ_{irr}. It is well known that the Hodge class λ, the boundaries δ_{irr} and δ_{i:S}, and the tautological classes ψ_{i} for 1 ≤ i ≤ n, freely generate Pic(\overline{M}_{g,n}). To simplify notation, on \overline{M}_{g,1} we set δ_{i} := δ_{i:{1}} for 1 ≤ i ≤ g − 1. We have the following result whose proof we omit (cf. [AC]):

Proposition 2.5. If j : \overline{M}_{i,1} → \overline{M}_{g} is the map obtained by attaching a general marked curve of genus g − i to the marked point of each genus i curve, then

j^{∗}(λ) = λ, j^{∗}(δ_{0}) = δ_{0}, j^{∗}(δ_{i}) = −ψ + δ_{2i−g} and j^{∗}(δ_{k}) = δ_{i−k} + δ_{i+k−g} for k ≠ 0, i.

(Here we make the convention δ_{k} := 0, for k < 0.)

Let π : \overline{M}_{g,1} → \overline{M}_{g} be the forgetful morphism. We will need the following results which follow essentially from [AC], §1:

Lemma 2.6. 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 2.7. 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. [EH3], Theorem 1). When g+1 is prime one has to use instead the class of the Petri divisor, which we do not reproduce here (cf. [EH3], 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 2.8. 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 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 B · j^{∗}(D) < 0, where B ⊂ \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 m ∈ ℤ and E is an effective divisor not containing the irreducible divisor π^{∗}(\overline{K}). From Proposition 2.5 we have that j^{∗}(λ) = λ, j^{∗}(δ_{0}) = δ_{0} and j^{∗}(δ_{10}) = −ψ, while \overline{K} ≡ 7λ−δ_{0} −⋯, hence

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

Moreover from Lemma 2.1 we obtain that R · π^{∗}(\overline{K}) = π_{∗}(R) · \overline{K} = −1, which yields the inequality

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

Since E is an effective class on \overline{M}_{10.1}, from Corollary 2.8 it follows that π_{∗}(\overline{W} · E) is an effective class on \overline{M}_{10}. An easy calculation using Lemma 2.6 shows that

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

We now use that for every effective divisor on \overline{M}_{g} the coefficient a of λ is nonnegative 3. From the previous formula we get an inequality which combined with (1) yields, after a simple computation

b_{10} ≥ 78 · \frac{1155}{1262} · b_{0} − 11 · \frac{1170}{1262} · a ( = (71.3866...) · b_{0} − (10.1980...) · a).

□

Question 2.9. Do we always have the inequality b_{10} ≥ 78b_{0} − 11a?

3For the reader’s convenience we recall that this follows from the fact that B ·λ > 0 for any irreducible 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 (outside of the given divisor). The inequality B · λ > 0 is a consequence of writing λ = (λ − ϵδ) + ϵδ where 0 < ϵ < 1/11 and of using that λ − ϵδ is ample on \overline{M}_{g} (cf. [CH]).

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. [EH3], 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 2.10. An amusing consequence of Corollary 2.8 is that the Kodaira dimension of the universal curve M_{g,1} is −∞ for all g ≤ 15, with g ≠ 13, 14 (this can be proved directly when g ≤ 11). Indeed, if we assume that some multiple of 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 Corollary 2.8 the same multiple of 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].

3. The four incarnations of the divisor \overline{K}

This section is devoted to the proof of Theorem 1.7. We start by reviewing some notation. 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 equipped with a natural rational map ϕ_{g} : P_{g} − − > M_{g}. By Mukai’s results [M1] this map is dominant if and only if 2 ≤ g ≤ 9 or g = 11. Moreover, ϕ_{g} is generically finite if and only if g = 11 or g ≥ 13. For g = 10 the map ϕ_{g} has fibre dimension 3, whereas the fibre dimension of ϕ_{12} is 1. The non-finiteness of ϕ_{g} is due to the existence of Fano threefolds X_{2g−2} ⊂ P^{g+1} of genus g = 10, 12.

We denote by \overline{K}_{g} the closure in M_{g} of the image of ϕ_{g}. Thus \overline{K} = \overline{K}_{10} is an irreducible divisor on M_{10}. By \overline{\overline{K}}_{g} we shall denote the Deligne-Mumford closure of \overline{K}_{g} in \overline{M}_{g}.

For a smooth curve C and a line bundle L on C we consider the Wahl map ψ_{L} : ∧^{2}H^{0}(L) → H^{0}(K_{C} ⊗ L^{⊗^{2}}) sending a pencil in the linear system |L| to its ramification divisor, that is,

ψ_{L}(σ ∧ τ) := σ dτ − τ dσ, for σ, τ ∈ H^{0}(L).

J. Wahl proved that if a smooth curve C sits on a K3 surface then ψ_{K} is not surjective (cf. [W]). This is the only known intrinsic characterization of curves sitting on a K3 surface. Later, Cukierman and Ulmer showed that for g = 10 the converse of Wahl’s theorem also holds (cf. [CU]). It is believed that for Brill-Noether-Petri general curves C of genus g ≥ 13 the non-surjectivity of ψ_{K} is equivalent to the existence of a K3 surface containing C. From now on we fix g = 10 and let C be a Brill-Noether general curve of genus 10. Then C carries finitely many base point free pencils 𝔤_{6}^{1}. The dual linear series 𝔤_{12}^{4} = |K_{C} − 𝔤_{6}^{1}| yield embeddings C ⊂ P^{4} with deg(C) = 12. We show that we can interpret points in the divisor \overline{K} in four geometrically meaningful ways:

Proof of Theorem 1.7. Since being Brill-Noether special is a condition of codimension 2 on \overline{M}_{10}, the general point of each component of the divisors on M_{10} defined by (1)−(4) will correspond to a Brill-Noether general curve. The fact that (1) and (2) are equivalent is the main result of [CU]. The implications (1) ⇒ (3) and (2) ⇒ (4) were proved by Voisin (cf. [V1], Proposition 3.2). We are left with showing that (3) ⇒ (4) and that (4) ⇒ (2).

(3)⇒(4). Let C be a Brill-Noether general curve of genus 10 and E ∈ SU(2, K_{C}) a vector bundle with h^{0}(E) ≥ 7. We denote by O_{C}(D) the maximal sub-line bundle of E. We have the exact sequence:

(2)  0 ⟶ O_{C}(D) ⟶ E ⟶ K_{C}(−D) ⟶ 0.

Using [PR] Lemma 3.9 we can assume that h^{0}(D) ≥ 2, hence deg(D) ≥ 6 (C is not 5-gonal). From the exact sequence and from Riemann-Roch we get that

7 ≤ h^{0}(E) ≤ h^{0}(D) + h^{0}(K_{C} − D) = 2h^{0}(D) − deg(D) + 9.

Again, since the curve is general, one can easily see that this can happen only when deg(D) = 6 and h^{0}(E) = h^{0}(D)+h^{0}(K_{C} −D), that is, when the cohomology coboundary map δ_{E} ∈ (H^{0}(K_{C} − D)^{∨})^{⊗^{2}} associated to (2) is zero. One can find a semi-stable bundle E ∈ Ext^{1}(K_{C}(−D), O_{C}(D)) = H^{0}(2K_{C} − 2D)^{∨} with δ_{E} = 0, precisely when the multiplication map

μ(D) : Sym^{2}H^{0}(K_{C} − D) → H^{0}(2K_{C} − 2D)

is not surjective. Since both vector spaces have dimension 15, this is equivalent to saying

that the image of the embedding C ⟶^{|K_{C}−D|} P^{4} sits on a quadric.

(4)⇒(2). This was shown to us by R. Lazarsfeld. Again, we fix C a Brill-Noether general curve of genus 10 and A ∈ W_{6}^{1}(C) such that the multiplication map μ(A) : Sym^{2}H^{0}(K_{C} − A) → H^{0}(2K_{C} − 2A) is not surjective. We get from Riemann-Roch that dim Sym^{2}H^{0}(K_{C} − A) = dim H^{0}(2K_{C} − 2A) = 15, which implies that μ(A) is not injective either. Let s_{1}, s_{2} ∈ H^{0}(A) be independent sections. One has the following commutative diagram:

Sym^{2}H^{0}(K_{C} − A) ⟶^{μ(A)} H^{0}(2K_{C} − 2A)
↓^{α}  ↓^{β}
∧^{2}H^{0}(K_{C})  ⟶^{ψ_{K}} H^{0}(C, 3K_{C})

Here α(ϵ · η) := ϵs_{1} ∧ ηs_{2} − ϵs_{2} ∧ ηs_{1} for ϵ, η ∈ H^{0}(K_{C} − A), while β(u) := (us_{1}) ds_{2} − (us_{2}) ds_{1}, for u ∈ H^{0}(2K_{C} − 2A). To conclude that ψ_{K} is not surjective (or equivalently injective, since dim ∧^{2} H^{0}(K_{C}) = dim H^{0}(3K_{C}) = 45), it suffices to prove that α is injective.

To achieve this we define the map

m := (P e(A) ⊗ Id_{H^{0}(K_{C})}) ∘ (Id_{H^{0}(K_{C}−A)} ⊗ f) : H^{0}(K_{C} − A)^{⊗^{2}} → H^{0}(K_{C})^{⊗^{2}},

where P e(A) : H^{0}(A) ⊗ H^{0}(K_{C} − A) → H^{0}(K_{C}) is the Petri map, while the map f : H^{0}(K_{C} − A) → H^{0}(A) ⊗ H^{0}(K_{C}) is defined by

f(η) := s_{1} ⊗ (ηs_{2}) − s_{2} ⊗ (ηs_{1}).

One can check that m(ϵ ⊗ η) = ϵs_{1} ⊗ ηs_{2} − ϵs_{2} ⊗ ηs_{1}, where ϵ, η ∈ H^{0}(K_{C} − A), which shows that

m(ϵ ⊗ η + η ⊗ ϵ) = ϵs_{1} ∧ ηs_{2} − ϵs_{2} ∧ ηs_{1}.

Thus we get that m(Sym^{2}H^{0}(K_{C} − A)) ⊂ ∧^{2}H^{0}(K_{C}), so α = m_{|_{Sym^{2}H^{0}(K_{C} −A)}}. It is

easy to prove that f is always injective, thus when P e(A) is injective, it follows that m is injective as well.

We are left with the case when P e(A) is not injective (this is a divisorial condition on M_{10} and cannot be ruled out by a dimension count). From the base-point-free pencil trick, Ker(P e(A)) is one-dimensional and is generated by the element s_{1} ⊗ (us_{2}) − s_{2} ⊗ (us_{1}), where 0 ≠ u ∈ H^{0}(K_{C} − 2A). Then one checks that

Im(Id_{H^{0}(K_{C}−A)} ⊗ f) ∩ (Ker(P e(A)) ⊗ Id_{H^{0}(K_{C})}) = 0

(inside H^{0}(K_{C} − A) ⊗ H^{0}(A) ⊗ H^{0}(K_{C})), which shows that m is injective in this case as well. □

Remark 3.1. Theorem 1.7 establishes only a set-theoretic equality between the loci in M_{10} described by the conditions (1) − (4), that is, we do not necessarily have equalities between the appropriate cycles with the multiplicities coming from the natural scheme structures. For instance if W denotes the locus of smooth curves with a non-surjective Wahl map viewed as a determinantal variety, then we have the equality of divisors W = 4K (cf. [CU]). This can be interpreted as saying that the corank of the Wahl map ψ_{K} is equal to 4 for a general [C] ∈ \overline{K}.

Remark 3.2. If we use description (3) of K, it turns out that for a general [C] ∈ \overline{K}, the rank two vector bundle E ∈ SU_{2}(K_{C}) with h^{0}(E) ≥ 7 is unique. More precisely, if C is a section of a K3 section S then E = E_{|_{C}}, where E is a rank two vector bundle on S, which is an elementary transformation of the trivial bundle H^{0}(C, A) ⊗ O_{S} along C. Remarkably, the vector bundle constructed in this way does not depend on the choice of the pencil A ∈ W_{6}^{1}(C) (cf. [V1]).

4. Limit linear series and degeneration of multiplication maps

The aim of this section is to understand the following situation: suppose {L_{b}}_{b∈B^{∗}} and {M_{b}}_{b∈B^{∗}} are two families of line bundles over a 1-dimensional family of smooth curves {X_{b}}_{b∈B^{∗}}, where B^{∗} = B − {b_{0}} and b_{0} ∈ B. We ask what happens to the multiplication map

μ_{b} : H^{0}(X_{b}, L_{b}) ⊗ H^{0}(X_{b}, M_{b}) ⟶ H^{0}(X_{b}, L_{b} ⊗ M_{b})

as X_{b} degenerates to a singular curve of compact type X_{b_{0}}? The answer will be given in terms of limit linear series. Everything in this section is contained (at least implicitly) in [EH1] and [EH4].

First we recall a few definitions. We fix a smooth curve C and a point p ∈ C. If l = (L, V) is a linear series 𝔤_{d}^{r} on C with L ∈ Pic^{d}(C) and V ⊂ H^{0}(L), then by ordering the finite set {ord_{p}(σ)}_{σ∈V} we obtain the vanishing sequence of l at p

a^{l}(p) : 0 ≤ a_{0}^{l}(p) < … < a_{r}^{l}(p) ≤ d.

The weight of p with respect to l is defined as w^{l}(p) := ∑_{i=0}^{r}(a_{i}^{l}(p) − i).

For line bundles L and M on C and for an element ρ ∈ H^{0}(L) ⊗ H^{0}(M), we write that ord_{p}(σ) ≥ k if ρ is in the linear span of the elements of the form σ ⊗ τ, with σ ∈ H^{0}(L), τ ∈ H^{0}(M) and such that ord_{p}(σ) + ord_{p}(τ) ≥ k.

Let μ_{L,M} : H^{0}(L) ⊗ H^{0}(M) ⟶ H^{0}(L ⊗ M) be the multiplication map and ρ ∈ Ker(μ_{L,M}). We shall often use the following simple fact: if {σ_{i}} ⊂ H^{0}(L) and {τ_{j}} ⊂ H^{0}(M) are bases of the spaces of global sections adapted to the point p ∈ C, that is, satisfying the conditions

ord_{p}(σ_{i}) = a_{i}^{L}(p) and ord_{p}(τ_{j}) = a_{j}^{M}(p) for all relevant i and j,

then there are distinct pairs of integers (i_{1}, j_{1}) ≠ (i_{2}, j_{2}) such that

ord_{p}(ρ) = ord_{p}(σ_{i_{1}}) + ord_{p}(τ_{j_{1}}) = ord_{p}(σ_{i_{2}}) + ord_{p}(τ_{j_{2}}).

Suppose now that π : X → B is a family of genus g curves over B = Spec(R), with R being a complete DVR with local parameter t, and let 0, η denote the special and the generic point of B respectively. Assume furthermore that X_{η} is smooth and that X_{0} is singular but of compact type. If L_{η} is a line bundle on X_{η} then, as explained in [EH1], there is a canonical way to associate to each component Y of X_{0} a line bundle L^{Y} on X such that deg_{Z}(L_{|_{Z}}^{Y}) = 0 for every component Z of X_{0} different from Y. We set L_{Y} := L_{|_{Y}}^{Y} which is a line bundle on the smooth curve Y.

We fix σ ∈ π_{∗}L_{η} a section on the generic fibre. We denote by α the smallest integer such that t^{α}σ ∈ π_{∗}L^{Y}, that is, t^{α}σ ∈ π_{∗}L^{Y} − tπ_{∗}L^{Y}. Then we set

σ^{Y} := t^{α}σ ∈ π_{∗}L^{Y} and σ_{Y} := σ_{|_{Y}}^{Y} ∈ H^{0}(Y, L_{Y}).

For a different component Z of the special fibre X_{0} meeting Y at a point p, we define similarly L^{Z}, L_{Z}, σ^{Z} and σ_{Z}. We have the following compatibility relation between σ_{Y} and σ_{Z} (cf. [EH1], Proposition 2.2):

Lemma 4.1. If we write σ^{Z} = t^{β}σ^{Y} ∈ π_{∗}L^{Z} for a unique integer β, then

deg(L_{Y}) − ord_{p}(σ_{Y}) ≤ β ≤ ord_{p}(σ_{Z}).

An immediate consequence of this is the inequality

ord_{p}(σ_{Y}) + ord_{p}(σ_{Z}) ≥ deg(L_{Y}) = deg(L_{Z}).

Assume from now on that we have two line bundles L_{η} and M_{η} on X_{η} and we choose an element ρ ∈ H^{0}(X_{η}, L_{η}) ⊗_{R_{η}} H^{0}(X_{η}, M_{η}). If Y and Z are components of X_{0} meeting at p as above, we define ρ^{Y} := t^{γ}ρ ∈ H^{0}(X, L^{Y}) ⊗_{R} H^{0}(X, M^{Y}), where γ is the minimal integer with this property. We have a similar definition for ρ^{Z} ∈ H^{0}(X, L^{Z}) ⊗_{R} H^{0}(X, M^{Z}). Between the sections ρ^{Y} and ρ^{Z} there is a relation

ρ^{Z} = t^{α}ρ^{Y}

for a uniquely determined integer α. To determine α we proceed as follows: we choose bases of sections {σ_{i} = σ_{i}^{Y} } for H^{0}(X, L^{Y}) and {τ_{j} = τ_{j}^{Y} } for H^{0}(X, M^{Y}) that are adapted to the point p ∈ Y ∩ Z, in the sense that

ord_{p}(σ_{i}) = a_{i}^{L_{Y}} (p) and ord_{p}(τ_{j}) = a_{j}^{M_{Y}} (p),

for all relevant i and j (cf. e.g. [EH1], Lemma 2.3, for the fact that this can be done). Then there are integers α_{i} and β_{j} defined by

σ_{i}^{Z} = t^{α_{i}}σ_{i} and τ_{j}^{Z} = t^{β_{j}}τ_{j}.

To obtain a formula for the integer α we write ρ^{Y} = ∑_{i,j} f_{ij}σ_{i} ⊗ τ_{j}, where f_{ij} ∈ R. We have the identity

ρ^{Z} = ∑_{i,j}(t^{α−α_{i}−β_{j}}f_{ij})(t^{α_{i}}σ_{i}) ⊗ (t^{β_{j}}τ_{j}),

from which we easily deduce that

α = max_{i,j}{α_{i} + β_{j} − ν(f_{ij})},

where ν denotes the valuation on R (see also [EH4], Lemma 3.2).

Lemma 4.2. With the above notations, if ρ_{Y} := ρ_{|_{Y}}^{Y} ∈ H^{0}(Y, L_{Y}) ⊗ H^{0}(Y, M_{Y}) and ρ_{Z} := ρ_{|_{Z}}^{Z} ∈ H^{0}(Z, L_{Z}) ⊗ H^{0}(Z, M_{Z}), then

ord_{p}(ρ_{Y}) + ord_{p}(ρ_{Z}) ≥ deg(L_{Y}) + deg(M_{Y}).

Proof. By definition, there exists a pair of indices (i_{1}, j_{1}) such that ν(f_{i_{1}j_{1}}) = 0 and

ord_{p}(ρ_{Y}) = ord_{p}(σ_{i_{1},Y}) + ord_{p}(σ_{j_{1},Y})

and clearly α ≥ α_{i_{1}} + β_{j_{1}}. To get an estimate on ord_{p}(ρ_{Z}) we only have to take into account the pairs of indices (i, j) for which α_{i} + β_{j} = α + ν(f_{ij}) ≥ α_{i_{1}} + β_{j_{1}}. For at least one such pair (i, j) we have that

ord_{p}(ρ_{Z}) = ord_{p}(t^{α_{i}}σ_{i,Z}) + ord_{p}(t^{β_{j}}τ_{j,Z}) ≥ α_{i} + β_{j}.

On the other hand, by applying Lemma 4.1 we can write

ord_{p}(ρ_{Y}) = ord_{p}(σ_{i_{1},Y}) + ord_{p}(τ_{j_{1},Y}) ≥ deg(L_{Y}) + deg(M_{Y}) − α_{i_{1}} − β_{j_{1}},

whence we finally have that ord_{p}(ρ_{Z}) + ord_{p}(ρ_{Y}) ≥ deg(L_{Y}) + deg(M_{Y}). □

5. The class of \overline{K}

We study in detail the Deligne-Mumford compactification of the K3 locus. Since Theorem 1.7 gives four different characterizations for the points of \overline{K}, to compute the class of \overline{K} in Pic(\overline{M}_{10}) one can try to understand what happens to each of the conditions (1) − (4) as a smooth genus 10 curve C degenerates to a singular stable curve.

It seems very difficult to understand the degenerations of [C] ∈ \overline{K} using directly characterization (1), although it would be highly interesting to have a description of all stable limits of K3 sections of genus g (perhaps as curves sitting on Kulikov degenerations of K3 surfaces – cf. also Remark 7.4). It also seems almost certain that one cannot use the Wahl map and description (2) to carry out any intersection theoretic computations on \overline{M}_{g}: although the Wahl map can be naturally extended to all stable nodal curves [C] ∈ \overline{M}_{g} as ψ_{ω_{C}} : ∧^{2}H^{0}(ω_{C}) → H^{0}(ω_{C}^{⊗^{2}} ⊗ K_{C}), where ω_{C} is the dualizing (locally free) sheaf of C and K_{C} = Ω_{C}^{1}, it is easy to see that as soon as C has a disconnecting node (in particular whenever C is of compact type), ψ_{ω_{C}} cannot be surjective for trivial reasons.

Instead, in order to understand \overline{K} we shall use description (4) from Theorem 1.7 together with the set-up developed in §4.

We recall some basic things about Brill-Noether divisors on the universal curve \overline{M}_{g,1}. Let us fix positive integers g, r, d and a ramification sequence α = (0 ≤ α_{0} ≤ … ≤ α_{r} ≤ d − r) such that

ρ(g, r, d) − w(α) = g − (r + 1)(g − d + r) − ∑_{i=0}^{r} α_{i} = −1.

We define M_{g,d}^{r}(α) to be the locus of pointed curves (C, p) ∈ M_{g,1} such that C carries a 𝔤_{d}^{r}, say l, with ramification at p at least α, that is, a_{i}^{l}(p) ≥ α_{i} + i, for i = 0, …, r. Eisenbud and Harris proved (cf. [EH2], Theorem 1.2) that the compactification \overline{M}_{g,d}^{r}(α) is a divisor on \overline{M}_{g,1} which we shall call a Brill-Noether divisor on \overline{M}_{g,1}. They also showed that its class is a linear combination

\overline{M}_{g,d}^{r}(α) ≡ a · BN_{1} + b · \overline{W}, with a, b ∈ ℚ_{≥0},

where

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

is the pull-back from \overline{M}_{g} of the Brill-Noether class and \overline{W} is the closure of the Weierstrass locus, considered in §2 (cf. [EH2], Theorem 4.1).

By Theorem 1.7 (4) we view \overline{K} as the locus of genus 10 curves C having a line bundle M ∈ W_{12}^{4}(C) such that the multiplication map

(3)  μ_{M} : Sym^{2}H^{0}(M) → H^{0}(M^{⊗^{2}}) is not surjective (⇔ injective).

The class of \overline{K} can be then written then as

\overline{K} ≡ Aλ − B_{0}δ_{0} − B_{1}δ_{1} − ⋯ − B_{5}δ_{5},

where A, B_{i} ≥ 0. Cukierman and Ulmer showed that A = 7 and B_{0} = 1 (cf. [CU], Proposition 3.5), while Theorem 1.1 gives the inequalities B_{1} ≥ 5 and B_{i} ≥ 11 − i, for i = 2, …, 5. To compute the coefficients B_{i} we have to interpret condition (3) when C is a stable curve of compact type. We consider the maps j_{i} : \overline{M}_{i,1} → \overline{M}_{10} as in Proposition 2.5 and compute the pullback of \overline{K}. We have the following result:

Theorem 5.1. Let j_{i} : \overline{M}_{i,1} → \overline{M}_{10} be the map obtained by attaching a fixed general curve of genus 10−i. Then for all 1 ≤ i ≤ 4 the pullback j_{i}^{∗}(\overline{K}) is a union of Brill-Noether divisors on \overline{M}_{i,1}, hence its class is a linear combination of \overline{W} and the Brill-Noether class pulled back from \overline{M}_{i}.

Remark 5.2. To make Theorem 5.1 more precise, we can show that j_{1}^{∗}(\overline{K}) = 0, that j_{2}^{∗}(\overline{K}) is supported on the Weierstrass divisor of \overline{M}_{2,1}, while j_{3}^{∗}(\overline{K}) is supported on the union of the Weierstrass divisor on \overline{M}_{3,1} and the hyperelliptic locus.

As a corollary, combining this with Proposition 2.5, we get all the coefficients but B_{5} in the expression of the class of \overline{K}, thus proving Theorem 1.6.

Proof of Theorem 5.1. We shall only describe the case of the map j_{4} : \overline{M}_{4,1} → \overline{M}_{10}, the remaining cases involving the maps j_{1}, j_{2} and j_{3} being similar and simpler. We assume that the conclusion of the Theorem already holds for j_{1}, j_{2} and j_{3}. Throughout the proof we consider a genus 10 curve X_{0} = C ∪_{p} Y, where (C, p) is a general pointed curve of genus 6 fixed once and for all, while (Y, p) is an arbitrary smooth pointed curve of genus 4. We shall prove that we can choose [C, p] ∈ M_{6,1} sufficiently general such that for all [Y, p] ∈ M_{4,1} outside the union of all Brill-Noether divisors, we have that [X_{0}] ∉ \overline{K}. Suppose by contradiction that [X_{0}] ∈ \overline{K} and let π : X → B be a 1-dimensional family with smooth genus 10 general fibre X_{t} sitting on a K3 surface and special fibre X′_{0} semistably equivalent to X_{0} and obtained from X_{0} by inserting a (possibly empty) chain of P^{1}’s at the node p. According to Theorem 1.7 (4), on a smooth curve X_{t} near X′_{0}, there exists a line bundle L_{t} ∈ W_{12}^{4}(X_{t}) such that the multiplication map

μ_{t} : Sym^{2}H^{0}(X_{t}, L_{t}) → H^{0}(X_{t}, L_{t}^{⊗^{2}})

is not injective. Take 0 ≠ ρ_{t} ∈ Ker(μ_{t}) and denote by

{l_{Y} = (L_{Y} , V_{Y} ⊂ H^{0}(L_{Y})), l_{C} = (L_{C}, V_{C} ⊂ H^{0}(L_{C}))}

the induced limit 𝔤_{12}^{4} on X_{0} obtained by restriction from the corresponding limit 𝔤_{12}^{4} on X′_{0}. From general facts about limit linear series we know that there is a 1 : 1 correspondence between limit 𝔤_{12}^{4}’s on X′_{0} and X_{0}, and so we may as well assume that X_{0} = X′_{0}. According to Section 3 we obtain elements

ρ_{Y} ∈ Ker{Sym^{2}(V_{Y}) → H^{0}(L_{Y}^{⊗^{2}})} and ρ_{C} ∈ Ker{Sym^{2}(V_{C}) → H^{0}(L_{C}^{⊗^{2}})},

such that ord_{p}(ρ_{Y}) + ord_{p}(ρ_{C}) ≥ 12 + 12 = 24 (cf. Lemma 4.2).

Our assumption that both (Y, p) and (C, p) are Brill-Noether general gives, using the additivity of the Brill-Noether number, that

(4)  ρ(6, 4, 12) − ∑_{i=0}^{4}(a_{i}^{l_{C}}(p) − i) = 0, and

(5)  ρ(4, 4, 12) − ∑_{i=0}^{4}(a_{i}^{l_{Y}} (p) − i) = 0.

Since a_{i}^{l_{C}}(p) + a_{4−i}^{l_{Y}}(p) ≥ 12 for i = 0, …, 4, equalities (4) and (5) yield

(6)  a_{i}^{l_{C}}(p) + a_{4−i}^{l_{Y}}(p) = 12, for i = 0, …, 4.

Moreover, we have the general fact (cf. [EH3], Proposition 1.2), that a Brill-Noether general pointed curve (Z, p) ∈ M_{g,1} carries a 𝔤_{d}^{r} with ramification ≥ (α_{0}, …, α_{r}) at the point p if and only if

(7)  g − ∑_{i=0}^{r} max{0, α_{i} + g − d + r} ≥ 0.

(This is a strengthening of the inequality

ρ(g, r, d) − w(p) = ρ(g, r, d) − ∑_{i=0}^{r} α_{i} = g − ∑_{i=0}^{r}(α_{i} + g − d + r) ≥ 0.)

Conditions (4), (5), (6) and (7) cut down the number of numerical possibilities for the ramification at p of the limit 𝔤_{12}^{4} on X_{0} to three. To simplify notations we set a_{i} := a_{i}^{l_{Y}} (p) and b_{i} := a_{i}^{l_{C}}(p) for i = 0, …, 4. We have three distinct numerical situations which we shall investigate separately:

(1) (a_{0}, a_{1}, a_{2}, a_{3}, a_{4}) = (4, 5, 6, 9, 10) and (b_{0}, b_{1}, b_{2}, b_{3}, b_{4}) = (2, 3, 6, 7, 8)

(2) (a_{0}, a_{1}, a_{2}, a_{3}, a_{4}) = (4, 5, 7, 8, 10) and (b_{0}, b_{1}, b_{2}, b_{3}, b_{4}) = (2, 4, 5, 7, 8)

(3) (a_{0}, a_{1}, a_{2}, a_{3}, a_{4}) = (4, 6, 7, 8, 9) and (b_{0}, b_{1}, b_{2}, b_{3}, b_{4}) = (3, 4, 5, 6, 8).

We first study case (1), which will serve as a model for (2) and (3).

Since ρ_{C} ∈ Ker{Sym^{2}H^{0}(C, L_{C}) → H^{0}(C, L_{C}^{⊗^{2}})}, there must be distinct pairs of indices (i_{1}, j_{1}) ≠ (i_{2}, j_{2}) such that (cf. §3)

ord_{p}(ρ_{C}) = b_{i_{1}} + b_{j_{1}} = b_{i_{2}} + b_{j_{2}}.

Similarly, for Y we obtain pairs (i′_{1}, j′_{1}) ≠ (i′_{2}, j′_{2}) such that

ord_{p}(ρ_{Y}) = a_{i′_{1}} + a_{j′_{1}} = a_{i′_{2}} + a_{j′_{2}} ≥ 24 − ord_{p}(ρ_{C}).

Clearly ord_{p}(ρ_{C}) ≥ 9 and case (1) breaks into two subcases:

1_{a}) ord_{p}(ρ_{C}) = 9, hence ord_{p}(ρ_{Y}) ≥ 15(= 5 + 10 = 6 + 9). This means that ρ_{Y} ∈ Sym^{2}H^{0}(Y, L_{Y} (−5p)), where E := L_{Y} (−5p) is a 𝔤_{7}^{3} with vanishing (0, 1, 4, 5) at p. We reach a contradiction by showing that the map Sym^{2}H^{0}(Y, E) → H^{0}(Y, E^{⊗^{2}}) is injective when (Y, p) is Brill-Noether general. This follows from the base-point-free pencil trick which, applied here, says that the map

H^{0}(Y, E(−4p)) ⊗ H^{0}(Y, E) → H^{0}(Y, E^{⊗^{2}}(−4p))

has kernel H^{0}(Y, O_{Y} (4p)) which is one-dimensional (p ∈ Y is not a Weierstrass point).

1_{b}) ord_{p}(ρ_{C}) = 10(= 3 + 7 = 2 + 8), hence ord_{p}(ρ_{Y}) ≥ 14(= 4 + 10 = 5 + 9). This case is more complicated since we cannot reach a contradiction by working with Y alone as we did in (1_{a}): by counting dimensions it turns out that the map Sym^{2}H^{0}(Y, L_{Y} (−4p)) → H^{0}(Y, L_{Y}^{⊗^{2}}(−8p)) cannot be injective, so potentially we could find an element ρ_{Y} ∈ Ker{Sym^{2}(V_{Y}) → H^{0}(Y, L_{Y}^{⊗^{2}})} with ord_{p}(ρ_{Y}) ≥ 14.

We turn to the genus 6 curve C instead, and we denote M := L_{C}(−2p). Thus M ∈ W_{10}^{4}(C) and a^{M}(p) = (0, 1, 4, 5, 6), and we know that there exists an element

γ ∈ Ker{Sym^{2}H^{0}(C, M) → H^{0}(C, M^{⊗^{2}})}

such that ord_{p}(γ) = 6(= 0 + 6 = 1 + 5) (here γ is obtained from ρ_{C} by subtracting the base locus of the linear series |L_{C}|). By degeneration methods we show that such a γ cannot exist when (C, p) ∈ M_{6,1} is suitably general.

From general Brill-Noether theory we know that on C there are finitely many line bundles M ∈ W_{10}^{4}(C) satisfying a^{M}(p) = (0, 1, 4, 5, 6). They are all of the form M = K_{C} ⊗ A^{∨} ⊗ O_{C}(4p), where A ∈ W_{4}^{1}(C). Since the Hurwitz scheme of coverings C →^{4:1} P^{1} with a genus 6 source curve is irreducible, it follows that the variety

{(C, p, M) : (C, p) ∈ M_{6,1}, M ∈ W_{10}^{4}(C) and a^{M}(p) ≥ (0, 1, 4, 5, 6)}

is irreducible as well. Therefore to show that γ as above cannot exist when (C, p) ∈ M_{6,1} is general, it will be enough to prove the following:

Claim: The general genus 6 pointed curve (C, p) with C ⊂ P^{4}, deg(C) = 10 and a(p) = a^{O_{C}(1)}(p) = (0, 1, 4, 5, 6) does not sit on a quadric Q ⊂ P^{4} with ord_{p}(Q) ≥ 6.

Proof of claim. By semicontinuity, it is enough to construct one single such embedded curve. We start with the smooth monomial curve Γ = ν(P^{1}), where ν : P^{1} → P^{4}, ν(t) = [1 : t : t^{4} : t^{5} : t^{6}]. We set p := ν(0) ∈ Γ and then a(p) = (0, 1, 4, 5, 6). If x_{0}, x_{1}, x_{2}, x_{3} and x_{4} are the coordinates on P^{4} then Γ is contained in three quadrics:

Q_{1} : x_{1}x_{2} − x_{0}x_{3} = 0, Q_{2} : x_{0}x_{4} − x_{1}x_{3} = 0 and Q_{3} : x_{2}x_{4} − x_{3}^{2} = 0.

Note that ord_{p}(Q_{1}) = 5 and ord_{p}(Q_{2}), ord_{p}(Q_{3}) ≥ 6. Take now a general hyperplane H ⊂ P^{4} and denote by {p_{1}, …, p_{6}} = H ∩ Γ. The points p_{1}, …, p_{6} will be in general position in H = P^{3} and let R ⊂ P^{3} be the unique twisted cubic passing through them. We choose an elliptic quartic E ⊂ P^{3} that passes through p_{1}, …, p_{6} and set C := Γ ∪ E. Thus C ⊂ P^{4} is a stable curve of genus 6 with deg(C) = 10 and vanishing a(p) = (0, 1, 4, 5, 6) at the smooth point p. To show that C can be smoothed in P^{4} while preserving the ramification at p it is enough to notice that A := ω_{C}(4p) ⊗ O_{C}(−1) is a 𝔤_{4}^{1} on C and use that every pencil on a stable curve can be smoothed to nearby smooth curves (cf. e.g. [EH1]). Alternatively one can prove this smoothing result by employing the methods from [HH].

The quadrics in P^{4} containing Γ with order ≥ 6 at p will be those in the pencil {λQ_{2} + νQ_{3}}_{[λ:ν]∈P^{1}}. We prove that no quadric in this pencil can contain a general elliptic quartic E ⊂ H that passes through p_{1}, …, p_{6}, which will settle the claim. This follows from a dimension count: the space of elliptic quartics E ⊂ H through p_{1}, …, p_{6} is 4-dimensional, while {B_{[λ:ν]} = H ∩ (λQ_{2} + μQ_{3} = 0)}_{[λ:ν]∈P^{1}} is a pencil of quadrics in H containing the twisted cubic R. It is immediate to see that there are only ∞^{2} elliptic quartics on each of the quadrics B_{[λ:ν]} in the pencil, which finishes the proof of the claim and finally takes care of case (1_{b}).

1_{c}) ord_{p}(ρ_{C}) ≥ 11, in which case ord_{p}(ρ_{C}) = 14(= 7 + 7 = 6 + 8), and this is impossible since the map Sym^{2}H^{0}(C, L_{C}(−6p)) → H^{0}(C, L_{C}^{⊗^{2}}(−12p)) is injective.

Cases (2) and (3) are similar to (1). For instance (3) breaks into the following subcases:

3_{a}) ord_{p}(ρ_{C}) = 8(= 3 + 5 = 4 + 4), which can be dismissed right away by looking at the ramification on Y (it would imply that ord_{p}(ρ_{Y}) ≥ 16(= 7 + 9 = 8 + 8) which is impossible).

3_{b}) ord_{p}(ρ_{C}) = 9(= 3 + 6 = 4 + 5), when also ord_{p}(ρ_{Y}) = 15(= 9 + 6 = 7 + 8). This case is ruled out by applying the base-point-free pencil trick on Y just like we did in case (1_{a}).

3_{c}) ord_{p}(ρ_{C}) ≥ 10. This case is similar to (1_{b}). We set M := L_{C}(−3p) ∈ W_{9}^{4}(C). Then we have the vanishing sequence a^{M}(p) = (0, 1, 2, 3, 5) and there exists an element γ ∈ Ker{Sym^{2}H^{0}(C, M) → H^{0}(C, M^{⊗^{2}})} with ord_{p}(γ) ≥ 4.

Riemann-Roch gives that M = K_{C} ⊗ A^{∨} ⊗ O_{C}(5p), where A ∈ W_{6}^{1}(C) is such that |A|(−5p) ≠ ∅. Since the Hurwitz scheme of coverings C →^{6:1} P^{1} with a genus 6 source curve and a ramification point p of index 5 is irreducible, again, it will be sufficient to construct a single smooth curve C ⊂ P^{4} of genus 6, deg(C) = 9 and having a point p ∈ C with a(p) = (0, 1, 2, 3, 5) and such that C does not lie on a quadric Q ⊂ P^{4} with ord_{p}(Q) ≥ 4.

To construct C, we start with a pointed elliptic curve (Γ, p) which we embed in P^{4} by the linear series |5p|. We choose a general hyperplane H ⊂ P^{4} and denote by {p_{1}, …, p_{5}} = Γ ∩ H. Then we fix a general elliptic quartic E ⊂ H through p_{1}, …, p_{5} and set C := Γ ∪ E. We have that C ⊂ P^{4} is a stable curve of genus 6 and degree 9 with vanishing a(p) = (0, 1, 2, 3, 5). There is a 5-dimensional family of quadrics in P^{4} containing Γ and a 3-dimensional subfamily of quadrics Q ⊂ P^{4} with ord_{p}(Γ) ≥ 4 (cf. [Hu], Proposition IV.2.1). A dimension count similar to the one in (1_{b}) establishes that none of these ∞^{3} quadrics will contain a general elliptic quartic E ⊂ H that passes through p_{1}, …, p_{5}.

Thus we have proved that there is [C, p] ∈ M_{6,1} such that for any [Y, p] ∈ M_{4,1} which is outside the Brill-Noether divisors, we have that [Y ∪_{p} C] ∉ \overline{K}. The same conclusion holds if [Y, p] is a general element of the divisor Δ_{0} on \overline{M}_{4,1} (the base-point-free pencil trick and the usual Brill-Noether theory used in the proof carries through to general irreducible nodal curves, (cf. [EH3], Theorem 1.1)). Therefore j_{4}^{∗}(\overline{K}) is a union of Brill-Noether divisors and (possibly) multiples of the boundary divisors Δ_{1}, Δ_{2} and Δ_{3}, hence we can write j_{4}^{∗}(\overline{K}) ≡ α\overline{W} + βBN_{1} + c_{1}Δ_{1} + c_{2}Δ_{2} + c_{3}Δ_{3}, for α, β, c_{1}, c_{2}, c_{3} ∈ ℚ_{≥0}.

Since we assume having already proved that j_{i}^{∗}(\overline{K}) is a union of Brill-Noether divisors for i = 1, 2, 3, we have that

\overline{K} ≡ 7λ − δ − 5δ_{1} − 9δ_{2} − 12δ_{3} − B_{4}δ_{4} − B_{5}δ_{5}.

By identification we obtain α = 7/5, β = 6/5, B_{4} = 14 and c_{1} = c_{2} = c_{3} = 0, which finishes the proof. Note that we have also proved Theorem 1.6 in the process. □

Remark 5.3. An easy calculation shows that for 6 ≤ i ≤ 9, j_{i}^{∗}(\overline{K}) is not a combination of Brill-Noether divisors on \overline{M}_{i,1}, a fact that can be also seen in the last part of the proof of Theorem 5.1. We have been unable to determine j_{5}^{∗}(\overline{K}).

6. The Slope Conjecture and the Iitaka dimension of the Brill-Noether linear system on \overline{M}_{g}

This section, which should be considered at least partially joint with Sean Keel, emphasizes another somewhat surprising use of curves on K3 surfaces, this time in genus 11. We note, based on results of Mukai, that on \overline{M}_{11} there exist “many strongly independent” effective divisors of minimal slope. This seems to contradict earlier beliefs; see below for a more precise formulation. We denote by \overline{F}_{g} the Baily-Borel compactification of F_{g} (see e.g. [Loo] for a general reference). For a ℚ-Cartier divisor D on a variety X we denote by κ(D) = κ(X, D) its Iitaka dimension.

Recall that Harris and Morrison have conjectured that s(D) ≥ 6 + 12/(g + 1) for any effective divisor D on \overline{M}_{g}, with equality when g + 1 is composite. In this case, the quantity 6 + 12/(g + 1) is the slope of the Brill-Noether divisors on \overline{M}_{g}. Harris and Morrison wondered whether all effective divisors of slope 6 + 12/(g + 1) should consist of curves having some special character (cf. [HM], p. 324). In this direction, they proved that on \overline{M}_{3} the only irreducible divisor of slope 9(= 6 + 12/(g + 1)) is the hyperelliptic locus \overline{M}_{3,2}^{1}, and that on \overline{M}_{5} the only irreducible divisor of slope 8 is the trigonal locus \overline{M}_{5,3}^{1}. Moreover, a standard argument involving pencils of plane curves shows that on \overline{M}_{8}, the Brill-Noether divisor \overline{M}_{8,7}^{2} is the only irreducible divisor of slope 22/3, while a slightly more involved argument using pencils of 4-gonal curves proves a similar conclusion for the 4-gonal divisor \overline{M}_{7,4}^{1} on \overline{M}_{7}. One way to rephrase these results is to say that for g = 3, 5, 7, 8, every effective ℚ-divisor D with s(D) = 6+12/(g +1) has Iitaka dimension κ(D) = 0, and each such D is an effective combination of the (unique) Brill-Noether divisor \overline{M}_{g,d}^{r} and boundary divisors Δ_{1}, …, Δ_{[g/2]}. In the case when g + 1 is multiply divisible and there are several Brill-Noether divisors \overline{M}_{g,d}^{r}, it is natural to ask the following:

Question 6.1. For composite g + 1, if a denotes the number of distinct Brill-Noether divisors on \overline{M}_{g}, is it true that κ(\overline{M}_{g}, D) ≤ a − 1 for any effective ℚ-divisor D on \overline{M}_{g} with slope s(D) = 6 + 12/(g + 1)?

The first interesting case is g = 11, when there are two distinct Brill-Noether divisors \overline{M}_{11,6}^{1} and \overline{M}_{11,9}^{2}. We are asking whether κ(BN + ∑_{i=1}^{5} a_{i}δ_{i}) = 1 for all a_{i} ≥ 0, i = 1, …, 5, where BN := 7λ − δ_{0} − 5δ_{1} − 9δ_{2} − 12δ_{3} − 14δ_{4} − 15δ_{5} is the Brill-Noether class. (The next interesting case is that of \overline{M}_{23}. This case was studied extensively in [F], and the Iitaka dimension of the Brill-Noether system is conjectured to give a positive answer to the question above.) We show that at least in this genus, the answer to Question 6.1 is far from being positive:

Proposition 6.2. There exist effective divisors on \overline{M}_{11} of slope 7 and having Iitaka dimension equal to 19. For example κ(\overline{M}_{11}, BN + 4δ_{3} + 7δ_{4} + 8δ_{5}) = 19.

Proof. By work of Mukai (cf. [M2]) there exists a rational map ϕ : \overline{M}_{11} − − > \overline{F}_{11} which sends a general curve [C] ∈ M_{11} to [S, C] ∈ F_{11}, where S is the unique K3 surface containing C. We denote by A the indeterminacy locus of ϕ. Note that since \overline{M}_{11} is normal, codim(A, \overline{M}_{11}) ≥ 2. For a ℚ-Cartier divisor D on \overline{F}_{11} we define the pull-back ϕ^{∗}D, for example as p_{∗}q^{∗}(D), where p : Σ → \overline{M}_{11} and q : Σ → \overline{F}_{11} are the projections from the closure of the graph of ϕ. It is easy to check that κ(\overline{M}_{11}, ϕ^{∗}D) ≥ κ(\overline{F}_{11}, D).

Take D now to be any ample effective divisor on \overline{F}_{11} (e.g. the zero locus of an automorphic form on the period space of K3 surfaces). Let us write ϕ^{∗}(D) ≡ aλ − ∑_{i=0}^{5} b_{i}δ_{i}, and we claim that a/b_{0} = 7. We choose a general [S, C] ∈ F_{11} such that Pic(S) = ℤC and S is the only K3 surface containing C. We also pick a Lefschetz pencil on S, giving rise to a curve B in \overline{M}_{11}. Since B fills-up \overline{M}_{11}, we can assume that B ∩ A = ∅ and that ϕ(b) = [S, C] for a general b ∈ B, hence ϕ(B) = {[S, C]}, that is, ϕ contracts the curve B. Then B · ϕ^{∗}(D) = 0. On the other hand we have that B · λ = g + 1 = 12, B · δ_{0} = 6(g + 3) = 84 and B · δ_{i} = 0 for i ≥ 1 (cf. Lemma 2.1), thus we obtain that a/b_{0} = B · δ_{0}/B · λ = 7.

We decompose ϕ^{∗}(D) into its moving and fixed part, ϕ^{∗}(D) = M + F, and choose a general member D′ ∈ |M|. Then D′ ≡ a′λ − ∑_{i=0}^{5} b′_{i}δ_{i} contains none of the boundaries Δ_{1}, …, Δ_{5} as components, and from Theorem 1.1 we obtain that a′/b′_{0} = a/b_{0} = 7, b′_{1} ≥ 5b′_{0} and b′_{i} ≥ (11 − i)b′_{0} for i = 2, …, 5. If we set E := BN + 4δ_{3} + 7δ_{4} + 8δ_{5}, it follows then that κ(\overline{M}_{11}, E) ≥ κ(\overline{M}_{11}, D′) = κ(\overline{M}_{11}, ϕ^{∗}(D)) ≥ 19.

On the other hand we claim that κ(\overline{M}_{11}, E) ≤ 19. Indeed, since B ·E = 0 we have that the rational map associated to any multiple of E contracts the 11-dimensional family of curves corresponding to the linear system |C| on S, thus κ(\overline{M}_{11}, E) ≤ dim(\overline{M}_{11})−11 = 19. □

7. Further applications

The K3 locus vs. Brill-Noether loci. For positive integers g, r, d such that ρ(g, r, d) < 0, a Brill-Noether locus M_{g,d}^{r} on M_{g} is defined as the locus of curves carrying a 𝔤_{d}^{r}, and we denote by \overline{M}_{g,d}^{r} its Deligne-Mumford compactification. When ρ(g, r, d) = −1, it is known that M_{g,d}^{r} is an irreducible divisor (cf. [EH2]). Virtually all of our knowledge about the effective cone of \overline{M}_{g} comes from the study of the Brill-Noether divisors \overline{M}_{g,d}^{r}. To emphasize the significant difference between \overline{K} and M_{g,d}^{r} as well as the potential for getting new information on the birational geometry of \overline{M}_{g} by studying higher genus analogues/generalizations of \overline{K}, we shall compare their behavior under the flag map ϕ : \overline{M}_{0,g} → \overline{M}_{g} obtained by attaching a fixed elliptic curve to the marked points of a g-pointed rational curve. A key result of modern Brill-Noether theory is the following:

Theorem 7.1. ([EH3], Theorem 1.1) Whenever ρ(g, r, d) < 0 we have that ϕ^{∗}(\overline{M}_{g,d}^{r}) = 0, that is, a flag curve of genus g with g elliptic tails is Brill-Noether general.

This result, besides offering a proof of the Brill-Noether Theorem, can be employed to compute almost all the coefficients of the class of \overline{M}_{g,d}^{r} when ρ(g, r, d) = −1 (cf. [EH3], Theorem 3.1). In contrast, for the locus of K3 sections \overline{K}_{g} we have the following result:

Proposition 7.2. The flag locus of genus g is entirely contained in \overline{K}_{g}, that is, ϕ(\overline{M}_{0,g}) ⊂ \overline{K}_{g}.

Proof. Let us denote by E the elliptic tail which appears in the definition of ϕ and denote by S the (K3) Kummer surface associated with E ×E. Pick a ∈ E[2] an element of order 2 and denote by R the strict transform of {a} × E. Then R is a smooth rational curve and R^{2} = −2. Pick also x_{1}, …, x_{g} ∈ E − E[2] arbitrary points and denote by C_{i} the strict transform of E × {x_{i}} for i = 1, …, g. Then all C_{i} are elliptic curves isomorphic to E such that C_{i} · R = 1, hence R + C_{1} + ⋯ + C_{g} is a flag curve of genus g sitting on the K3 surface S. □

Theorem 1.6 gives detailed information about degenerate K3 sections, that is, stable curves that are limits of smooth K3 sections. For instance one can prove the following:

Proposition 7.3. Every stable curve of genus 10 with five tails of genus two is a degenerate K3 section. Every genus 10 curve with one elliptic and three genus three tails is a degenerate K3 section.

Proof. We only consider the first case. We look at the map m : \overline{M}_{0,5} → \overline{M}_{10} obtained by attaching five arbitrary genus two tails at the marked points x_{1}, …, x_{5} of each element from \overline{M}_{0,5}. If B_{2} denotes the boundary divisor of \overline{M}_{0,5} corresponding to singular rational curves of type (2, 3), then it is easy to see that

m^{∗}(δ_{4}) = B_{2}, m^{∗}(δ_{2}) = − ∑_{i=1}^{5} ψ_{x_{i}} = −\frac{3}{2}B_{2}, m^{∗}(λ) = m^{∗}(δ_{i}) = 0 for i ∈ \{0, 1, 3, 5\}.

Thus m^{∗}(\overline{K}) = −\frac{3}{2}B_{2} and so Im(m) ⊂ \overline{K}. □

Remark 7.4. It would be interesting to realize explicitly such genus 10 curves with five genus two tails as sections of some K3 surfaces. Such a surface will necessarily be a degenerate one. In fact, if C is a singular genus g curve of compact type sitting on a smooth K3 surface S, then using well known facts about linear systems on K3 surfaces, one can prove that S is elliptic and C consists only of rational and elliptic curves.

The Kodaira dimension of \overline{M}_{10,n}. It is known that for each g ≥ 3, there is an integer f(g) such that \overline{M}_{g,n} is of general type for all n ≥ f(g) (cf. [Log], Theorem 2.4). For those values of g for which M_{g} is unirational (or more generally κ(\overline{M}_{g}) = −∞), a natural question to ask is to determine f(g). We show that Theorem 1.6 can be used to give an answer to this question for g = 10. Recall first that one has the formula for the canonical class (cf. [Log], Theorem 2.6)

K_{\overline{M}_{g,n}} ≡ 13λ − 2δ_{irr} + ∑_{i=1}^{n} ψ_{i} − 2 ∑_{i≥0,S} δ_{i:S} − ∑_{S} δ_{1:S}.

To prove that K_{\overline{M}_{g,n}} is effective for certain g and n we are going to use besides the divisor \overline{K}, the effective divisor D on \overline{M}_{g,g} consisting of pointed genus g curves (C, p_{1}, …, p_{g}) such that h^{0}(C, p_{1} + ⋯ + p_{g}) ≥ 2. The class of D has been computed in [Log], Theorem 5.4, and we have the formula

D ≡ −λ + ∑_{i=1}^{g} ψ_{i} − ∑_{i≥0,S} c_{i:S} δ_{i:S},

where c_{0:S} ≥ 2 and c_{i:S} > 0, for all relevant i and S (note that the coefficient of δ_{irr} is 0).

Proposition 7.5. The Kodaira dimension of \overline{M}_{10,10} is ≥ 0. For all n ≥ 11 we have that \overline{M}_{10,n} is of general type.

Proof. We fix an integer n ≥ 10 and denote by π_{n} : \overline{M}_{10,n} → \overline{M}_{10} the morphism forgetting all the marked points. We consider two effective divisors on \overline{M}_{10,n}: firstly, the pullback of the K3 locus

𝔎_{n} := π_{n}^{∗}(\overline{K}) ≡ 7λ − δ_{irr} − 5 ∑_{S} δ_{1:S} − 9 ∑_{S} δ_{2:S} − 12 ∑_{S} δ_{3:S} − 14 ∑_{S} δ_{4:S} − B_{5} ∑_{S} δ_{5:S}.

Secondly, if for any S ⊂ \{1, …, n\} with |S| = 10, we denote by π_{S} : \overline{M}_{10,n} → \overline{M}_{10,10} the morphism forgetting the marked points from S^{c}, by averaging the pullbacks of D, we define the effective divisor on \overline{M}_{10,n}

𝔇_{n} := \frac{1}{\binom{n}{10}} ∑_{|S|=10} π_{S}^{∗}(D) ≡ −λ + \frac{10}{n} ∑_{i=1}^{n} ψ_{i} − ∑_{i≥0,S} c̃_{i:S} δ_{i:S},

where c̃_{0:S} ≥ 2 and c̃_{i:S} > 0. A check yields that K_{\overline{M}_{10,10}} − 2𝔎_{10} − 𝔇_{10} is an effective combination of boundary classes δ_{i:S}, thus proving that κ(\overline{M}_{10,10}) ≥ 0. Similarly, for n ≥ 11, one can find positive constants α, β, a, b_{irr} and c_{i} for 1 ≤ i ≤ n, with a/b_{irr} arbitrarily large, such that K_{\overline{M}_{10,n}} −α𝔎_{n} −β𝔇_{n} −(aλ−b_{irr}δ_{irr} +∑_{i=1}^{n} c_{i}ψ_{i}) is a positive combination of boundary classes δ_{i:S}. As the class aλ−b_{irr}δ_{irr} +∑_{i=1}^{n} c_{i}ψ_{i} is big for such a choice (cf. e.g. [Log], Theorem 2.9), this implies that \overline{M}_{10,n} is of general type. □

Using again the divisor \overline{K} we can show that Proposition 7.5 is optimal:

Proposition 7.6. The Kodaira dimension of \overline{M}_{10,n} is −∞ for n ≤ 9.

Proof. We are only going to prove this for n = 9, which will imply the same conclusion for lower n. We consider the divisor 𝔎_{9} = π_{9}^{∗}(\overline{K}) on \overline{M}_{10,9} and a general point [C, p_{1}, …, p_{9}] ∈ 𝔎_{9} corresponding to a curve C sitting on a K3 surface S. Since dim|O_{S}(C)| = 10, from the generality of C ⊂ S and of the points p_{i} ∈ C, it follows that |O_{S}(C) ⊗ I_{\{p_{1},...,p_{9}}\}| is a pencil giving rise to a curve R ⊂ \overline{M}_{10,9}. One finds that R · λ = 11, R · δ_{irr} = 78, R · ψ_{i} = 1 for all 1 ≤ i ≤ 9 while R · δ_{i:S} = 0 for all i and S (cf. Lemma 2.4 and completely similar calculations). It follows that R · K_{\overline{M}_{10,9}} = −4.

Next we define a second curve T ⊂ Δ_{irr} ⊂ \overline{M}_{10,9}, obtained from a general pointed curve [Y, p, p_{1}, …, p_{9}] ∈ \overline{M}_{9,10} by identifying the fixed point p with a moving point y ∈ Y. A standard calculation shows that T · λ = 0, T · δ_{irr} = −2g(Y) = −18, T · δ_{1:∅} = 1 and T · ψ_{i} = 1 for 1 ≤ i ≤ 9, while T stays away from all other boundary divisors on \overline{M}_{10,9}.

We assume that K_{\overline{M}_{10,9}} is effective and write K_{\overline{M}_{10,9}} ≡ m · 𝔎_{9} + n · Δ_{irr} + E for m, n ∈ ℤ_{≥0}, where E is an effective divisor that contains neither 𝔎_{9} nor Δ_{irr}. Since the curves of the type of R fill-up the divisor 𝔎_{9}, while those of the type of T fill-up Δ_{irr}, we must have that R · E ≥ 0 and T · E ≥ 0. A direct check shows that these inequalities are not compatible with the conditions m, n ≥ 0. □

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Department of Mathematics, Princeton University, Fine Hall, Princeton, NJ 08544

      E-mail address: gfarkas@math.princeton.edu

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

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