The Geometry of the Moduli Space of Curves of Genus 23

GAVRIL FARKAS

1 Introduction

The problem of describing the birational geometry of the moduli space M_{g} of complex curves of genus g has a long history. Severi already knew in 1915 that M_{g} is unirational for g ≤ 10 (cf. [Sev]; see also [AC1] for a modern proof). In the same paper Severi conjectured that M_{g} is unirational for all genera g. Then for a long period this problem seemed intractable (Mumford writes in [Mu], p.51:“Whether more M_{g}’s, g ≥ 11, are unirational or not is a very interesting problem, but one which looks very hard too, especially if g is quite large”). The breakthrough came in the eighties when Eisenbud, Harris and Mumford proved that M_{g} is of general type as soon as g ≥ 24 and that the Kodaira dimension of M_{23} is ≥ 1 (see [HM], [EH3]). We note that M_{g} is rational for g ≤ 6 (see [Dol] for problems concerning the rationality of various moduli spaces).

Severi’s proof of the unirationality of M_{g} for small g was based on representing a general curve of genus g as a plane curve of degree d with δ nodes; this is possible when d ≥ 2g/3 + 2. When the number of nodes is small, i.e. δ < (d + 1)(d + 2)/6, the dominant map from the variety of plane curves of degree d and genus g to M_{g} yields a rational parametrization of the moduli space. The two conditions involving d and δ can be satisfied only when g ≤ 10, so Severi’s argument cannot be extended for other genera. However, using much more subtle ideas, Chang, Ran and Sernesi proved the unirationality of M_{g} for g = 11, 12, 13 (see [CR1], [Se1]), while for g = 15, 16 they proved that the Kodaira dimension is −∞ (see [CR2,4] ). The remaining cases g = 14 and 17 ≤ g ≤ 23 are still quite mysterious. Harris and Morrison conjectured in [HMo] that M_{g} is uniruled precisely when g < 23.

All these facts indicate that M_{23} is a very interesting transition case. Our main result is the following:

Theorem 1 The Kodaira dimension of the moduli space of curves of genus 23 is ≥ 2.

We will also present some evidence for the hypothesis that the Kodaira dimension of M_{23} is actually equal to 2. Acknowledgments: I am grateful to my advisor Gerard van der Geer, for proposing the problem and for continuous support and guidance. I thank Joe Harris for many inspiring discussions during my stay at Harvard. I also benefitted from conversations with Marc Coppens and Carel Faber.

2 Multicanonical linear systems and the Kodaira dimension of M_{g}

We study three multicanonical divisors on M_{23}, which are (modulo some boundary components) of Brill-Noether type and we conclude by looking at their relative position that κ(M_{23}) ≥ 2.

We review some notations. We shall denote by \overline{M}_{g} and \overline{C}_{g} the moduli spaces of stable and 1-pointed stable curves of genus g over ℂ. If C is a smooth algebraic curve of genus g, we consider for any r and d, the scheme whose points are the 𝔤_{d}^{r}’s on C, that is,

G_{d}^{r}(C) = {(L, V) : L ∈ Pic ^{d}(C), V ⊆ H^{0}(C, L), dim(V) = r + 1},

(cf. [ACGH]) and denote the associated Brill-Noether locus in M_{g} by

M_{g,d}^{r} := {[C] ∈ M_{g} : G_{d}^{r}(C) ≠ ∅},

and by \overline{M}_{g,d}^{r} its closure in \overline{M}_{g}.

The distribution of linear series on algebraic curves is governed (to some extent) by the Brill-Noether number

ρ(g, r, d) := g − (r + 1)(g − d + r).

The Brill-Noether Theorem asserts that when ρ(g, r, d) ≥ 0 every curve of genus g possesses a 𝔤_{d}^{r}, while when ρ(g, r, d) < 0 the general curve of genus g has no 𝔤_{d}^{r}’s, hence in this case the Brill-Noether loci are proper subvarieties of M_{g}. When ρ(g, r, d) < 0, the naive expectation that −ρ(g, r, d) is the codimension of M_{g,d}^{r} inside M_{g}, is in general way off the mark, since there are plenty of examples of Brill-Noether loci of unexpected dimension (cf. [EH2]). However, we have Steffen’s result in one direction (see [St]):

If ρ(g, r, d) < 0 then each component of M_{g,d}^{r} has codimension at most −ρ(g, r, d) in M_{g}.

On the other hand, when the Brill-Noether number is not very negative, the Brill-Noether loci tend to behave nicely. Existence of components of M_{g,d}^{r} of the expected dimension has been proved for a rather wide range (cf. [EH1]), namely for those g, r, d such that ρ(g, r, d) < 0, and

ρ(g, r, d) ≥ \begin{cases}−g + r + 3 & if r is odd; \\ −rg/(r + 2) + r + 3 & if r is even. \end{cases}

We have a complete answer only when ρ(g, r, d) = −1. Eisenbud and Harris have proved in [EH2] that in this case M_{g,d}^{r} has a unique divisorial component, and using the previously mentioned theorem of Steffen’s, we obtain the following result:

If ρ(g, r, d) = −1, then \overline{M}_{g,d}^{r} is an irreducible divisor of \overline{M}_{g}.

We will also need Edidin’s result (see [Ed2] ) which says that for g ≥ 12 and ρ(g, r, d) = −2, all components of M_{g,d}^{r} have codimension 2. We can get codimension 1 Brill-Noether conditions only for the genera g for which g + 1 is composite. In that case we can write

g + 1 = (r + 1)(s − 1), s ≥ 3

and set d := rs − 1. Obviously ρ(g, r, d) = −1 and \overline{M}_{g,d}^{r} is an irreducible divisor. Furthermore, its class has been computed (cf. [EH3] ):

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

where c_{g,r,d} is a positive rational number equal to 3μ/(2g −4), with μ being the number of 𝔤_{d}^{r}’s on a general pointed curve (C_{0}, q) of genus g − 2 with ramification sequence (0, 1, 2, …, 2) at q. For g = 23 we have the following possibilities:

(r, s, d) = (1, 13, 12), (11, 3, 32), (2, 9, 17), (7, 4, 27), (3, 7, 20), (5, 5, 24).

It is immediate by Serre duality, that cases (1, 13, 12) and (11, 3, 32) yield the same divisor on M_{23}, namely the 12-gonal locus M_{12}^{1}; similarly, cases (2, 9, 17) and (7, 4, 27) yield the divisor M_{17}^{2} of curves having a 𝔤_{17}^{2}, while cases (3, 7, 20) and (5, 5, 24) give rise to M_{20}^{3}, the divisor of curves having a 𝔤_{20}^{3}. Note that when the genus we are referring to is clear from the context, we write M_{d}^{r} = M_{g,d}^{r}.

By comparing the classes of the Brill-Noether divisors to the class of the canonical divisor K_{\overline{M}_{g,reg}} = 13λ − 2δ_{0} − 3δ_{1} − 2δ_{2} − ⋯ − 2δ_{[g/2]}, at least in the case when g + 1 is composite we can infer that

K_{\overline{M}_{g,reg}} = a[\overline{M}_{g,d}^{r}] + bλ + ( positive combination of δ_{0}, …, δ_{[g/2]}),

where a is a positive rational number, while b > 0 as long as g ≥ 24 but b = 0 for g = 23. As it is well-known that λ is big on \overline{M}_{g}, it follows that M_{g} is of general type for g ≥ 24 and that it has non-negative Kodaira dimension when g = 23. Specifically for g = 23, we get that there are positive integer constants m, m_{1}, m_{2}, m_{3} such that:

mK = m_{1}[\overline{M}_{12}^{1}] + E, mK = m_{2}[\overline{M}_{17}^{2}] + E, mK = m_{3}[\overline{M}_{20}^{3}] + E,  (1)

where E is the same positive combination of δ_{1}, …, δ_{11}.

Proposition 2.1 (Eisenbud-Harris, [EH3]) There exists a smooth curve of genus 23 that possesses a 𝔤_{12}^{1}, but no 𝔤_{17}^{2}. It follows that κ(M_{23}) ≥ 1.

Harris and Mumford proved (cf. [HM]) that \overline{M}_{g} has only canonical singularities for g ≥ 4, hence H^{0}(\overline{M}_{g,reg}, nK) = H^{0}(M̃_{g}, nK) for each n ≥ 0, with M̃_{g} a desingularization of \overline{M}_{g}. We already know that dim(Imϕ_{mK}) ≥ 1, where ϕ_{mK} : \overline{M}_{23} −− → ℙ^{ν} is the multicanonical map, m being as in (1). We will prove that κ(M_{23}) ≥ 2. Indeed, let us assume that dim(Imϕ_{mK}) = 1. Denote by C := \overline{Imϕ_{mK}} the Kodaira image of \overline{M}_{23}.

We reach a contradiction by proving two things:

• α) The Brill-Noether divisors M_{12}^{1}, M_{17}^{2} and M_{20}^{3} are mutually distinct.

• β) There exist smooth curves of genus 23 which belong to exactly two of the Brill-Noether divisors from above. This suffices in order to prove Theorem 1: since \overline{M}_{12}^{1}, \overline{M}_{17}^{2} and \overline{M}_{20}^{3} are part of different multicanonical divisors, they must be contained in different fibres of the multicanonical map ϕ_{mK}. Hence there exists different points x, y, z ∈ C such that

M_{12}^{1} = \overline{ϕ^{−1}(x)} ∩ M_{23}, M_{17}^{2} = \overline{ϕ^{−1}(y)} ∩ M_{23}, M_{20}^{3} = \overline{ϕ^{−1}(z)} ∩ M_{23}.

It follows that the set-theoretic intersection of any two of them will be contained in the base locus of |mK_{\overline{M}_{23}}|. In particular:

supp(M_{12}^{1}) ∩ supp(M_{17}^{2}) = supp(M_{17}^{2}) ∩ supp(M_{20}^{3}) = supp(M_{20}^{3}) ∩ supp(M_{12}^{1}), (2)

and this contradicts β). We complete the proof of α) and β) in Section 5.

3 Deformation theory for 𝔤_{d}^{r}’s and limit linear series

We recall a few things about the variety parametrising 𝔤_{d}^{r}’s on the fibres of the universal curve (cf. [AC2]), and then we recap on the theory of limit linear series (cf. [EH1], [Mod]), which is our main technique for the study of M_{23}.

Given g, r, d and a point [C] ∈ M_{g}, there is a connected neighbourhood U of [C], a finite ramified covering h : M → U, such that M is a fine moduli space of curves (i.e. there exists ξ : C → M a universal curve), and a proper variety over M,

π : G_{d}^{r} → M

which parametrizes classes of couples (C, l), with [C] ∈ M and l ∈ G_{d}^{r}(C), where we have made the identification C = ξ^{−1}([C]) . Let (C, l) be a point of G_{d}^{r} corresponding to a curve C and a linear series l = (L, V), where L ∈ Pic^{d}(C), V ⊆ H^{0}(C, L), and dim(V) = r + 1. By choosing a basis in V, one has a morphism f : C → ℙ^{r}. The normal sheaf of f is defined through the exact sequence

0 ⟶ T_{C} ⟶ f^{∗}(T_{ℙ^{r}}) ⟶ N_{f} ⟶ 0.  (3)

By dividing out the torsion of N_{f} one gets to the exact sequence

0 ⟶ K_{f} ⟶ N_{f} ⟶ N′_{f} ⟶ 0,  (4)

where the torsion sheaf K_{f} (the cuspidal sheaf) is based at those points x ∈ C where df(x) = 0, and N′_{f} is locally free of rank r − 1. The tangent space T_{(C,l)}(G_{d}^{r}) fits into an exact sequence (cf. [AC2]):

0 ⟶ ℂ ⟶ Hom(V, V) ⟶ H^{0}(C, N_{f}) ⟶ T_{(C,l)}(G_{d}^{r}) ⟶ 0,  (5)

from which we have that dim T_{(C,l)}(G_{d}^{r}) = 3g − 3 + ρ(g, r, d) + h^{1}(C, N_{f}).

Proposition 3.1 Let C be a curve and l ∈ G_{d}^{r}(C) a base point free linear series. Then the variety G_{d}^{r} is smooth and of dimension 3g − 3 + ρ(g, r, d) at the point (C, l) if and only if H^{1}(C, N_{f}) = 0.

Remark: The condition H^{1}(C, N_{f}) = 0 is automatically satisfied for r = 1 as N_{f} is a sheaf with finite support. Thus G_{d}^{1} is smooth of dimension 2g + 2d − 5. It follows that G_{d}^{1} is birationally equivalent to the d-gonal locus M_{d}^{1} when d < (g + 2)/2.

Limit linear series try to answer questions of the following kind: what happens to a family of 𝔤_{d}^{r}’s when a smooth curve specializes to a reducible curve? Limit linear series solve such problems for a class of reducible curves, those of compact type. A curve C is of compact type if its dual graph is a tree. A curve C is tree-like if, after deleting edges leading from a node to itself, the dual graph becomes a tree.

Let C be a smooth curve of genus g and l = (L, V) ∈ G_{d}^{r}(C), L ∈ Pic^{d}(C), V ⊆ H^{0}(C, L), and dim(V) = r + 1. Fix p ∈ C a point. By ordering the finite set {ord_{p}(σ)}_{σ∈V} one gets the vanishing sequence of l at p:

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

The ramification sequence of l at p

α^{l}(p) : 0 ≤ α_{0}^{l}(p) ≤ … ≤ α_{r}^{l}(p) ≤ d − r

is defined as α_{i}^{l}(p) = a_{i}^{l}(p) − i and the weight of l at p is

w^{l}(p) = ∑_{i=0}^{r} α_{i}^{l}(p).

A Schubert index of type (r, d) is a sequence of integers β : 0 ≤ β_{0} ≤ … β_{r} ≤ d − r. If α and β are Schubert indices of type (r, d) we write α ≤ β ⟺ α_{i} ≤ β_{i}, i = 0, …, r. The point p is said to be a ramification point of l if w^{l}(p) > 0. The linear series l is said to have a cusp at p if α^{l}(p) ≥ (0, 1, …, 1). For C a tree-like curve, p_{1}, …, p_{n} ∈ C smooth points and α^{1}, …, α^{n} Schubert indices of type (r, d), we define

G_{d}^{r}(C, (p_{1}, α^{1}), … (p_{n}, α^{n})) := {l ∈ G_{d}^{r}(C) : α^{l}(p_{1}) ≥ α^{1}, …, α^{l}(p_{n}) ≥ α^{n}}.

This scheme can be realized naturally as a determinantal variety and its expected dimension is

ρ(g, r, d, α^{1}, …, α^{n}) := ρ(g, r, d) − ∑_{i=1}^{n} ∑_{j=0}^{r} α_{j}^{i}.

If C is a curve of compact type, a crude limit 𝔤_{d}^{r} on C is a collection of ordinary linear series l = {l_{Y} ∈ G_{d}^{r}(Y) : Y ⊆ C is a component}, satisfying the following compatibility condition: if Y and Z are components of C with {p} = Y ∩ Z, then

a_{i}^{l_{Y}} (p) + a_{r−i}^{l_{Z}}(p) ≥ d, for i = 0, … r.

If equality holds everywhere, we say that l is a refined limit 𝔤_{d}^{r}. The ‘honest’ linear series l_{Y} ∈ G_{d}^{r}(Y) is called the Y -aspect of the limit linear series l.

We will often use the additivity of the Brill-Noether number: if C is a curve of compact type, for each component Y ⊆ C, let q_{1}, …, q_{s} be the points where Y meets the other components of C. Then for any limit 𝔤_{d}^{r} on C we have the following inequality:

ρ(g, r, d) ≥ ∑_{Y ⊆C} ρ(l_{Y} , α^{l_{Y}} (q_{1}), …, α^{l_{Y}} (q_{s})),  (6)

with equality if and only if l is a refined limit linear series.

It has been proved in [EH1] that limit linear series arise indeed as limits of ordinary linear series on smooth curves. Suppose we are given a family π : C → B of genus g curves, where B = Spec(R) with R a complete discrete valuation ring. Assume furthermore that C is a smooth surface and that if 0, η denote the special and generic point of B respectively, the central fibre C_{0} is reduced and of compact type, while the generic geometric fibre C_{η} is smooth and irreducible. If l_{η} = (L_{η}, V_{η}) is a 𝔤_{d}^{r} on C_{η}, there is a canonical way to associate a crude limit series l_{0} on C_{0} which is the limit of l_{η} in a natural way: for each component Y of C_{0}, there exists a unique line bundle L^{Y} on C such that

L_{|C_{η}}^{Y} = L_{η} and deg_{Z}(L_{|_{Z}}^{Y}) = 0,

for any component Z of C_{0} with Z ≠ Y. (This implies of course that deg_{Y} (L_{|_{Y}}^{Y}) = d). Define V^{Y} = V_{η} ∩ H^{0}(C, L^{Y}) ⊆ H^{0}(C_{η}, L_{η}). Clearly, V^{Y} is a free R-module of rank r + 1. Moreover, the composite homomorphism

V^{Y} (0) → (π_{∗}L^{Y})(0) → H^{0}(C_{0}, L_{|_{C_{0}}}^{Y}) → H^{0}(Y, L_{|_{Y}}^{Y})

is injective, hence l_{Y} = (L_{|_{Y}}^{Y} , V^{Y} (0)) is an ordinary 𝔤_{d}^{r} on Y. One proves that l = {l_{Y} : Y component of C_{0}} is a limit linear series.

If C is a reducible curve of compact type, l a limit 𝔤_{d}^{r} on C, we say that l is smoothable if there exists π : C → B a family of curves with central fibre C = C_{0} as above, and (L_{η}, V_{η}) a 𝔤_{d}^{r} on the generic fibre C_{η} whose limit on C (in the sense previously described) is l. Remark: If a stable curve of compact type C, has no limit 𝔤_{d}^{r}’s, then [C] ∉ \overline{M}_{g,d}^{r}. If there exists a smoothable limit 𝔤_{d}^{r} on C, then [C] ∈ \overline{M}_{g,d}^{r}.

Now we explain a criterion due to Eisenbud and Harris (cf. [EH1]), which gives a sufficient condition for a limit 𝔤_{d}^{r} to be smoothable. Let l be a limit 𝔤_{d}^{r} on a curve C of compact type. Fix Y ⊆ C a component, and \{q_{1}, …, q_{s}\} = Y ∩ \overline{(C − Y)}. Let

π : Y → B, q̃_{i} : B → Y

be the versal deformation space of (Y, q_{1}, … q_{s}). The base B can be viewed as a small (3g(Y) − 3 + s)-dimensional polydisk. Using general theory one constructs a proper scheme over B,

σ : G_{d}^{r}(Y/B; (q̃_{i}, α^{l_{Y}} (q_{i}))_{i=1}^{s}) → B

whose fibre over each b ∈ B is σ^{−1}(b) = G_{d}^{r}(Y_{b}, (q̃_{i}(b), α^{l_{Y}} (q_{i}))_{i=1}^{s}). One says that l is dimensionally proper with respect to Y, if the Y -aspect l_{Y} is contained in some component G of G_{d}^{r}(Y/B; (q̃_{i}, α^{l_{Y}} (q_{i}))_{i=1}^{s}) of the expected dimension, i.e.

dim G = dim B + ρ(l_{Y} , α^{l_{Y}} (q_{1}), … α^{l_{Y}} (q_{s})).

One says that l is dimensionally proper, if it is dimensionally proper with respect to any component Y ⊆ C. The ‘Regeneration Theorem’ (cf. [EH1]) states that every dimensionally proper limit linear series is smoothable.

The next result is a ‘strong Brill-Noether Theorem’, i.e. it not only asserts a Brill-Noether type statement, but also singles out the locus where the statement fails.

Proposition 3.2 (Eisenbud-Harris) Let C be a tree-like curve and for any component Y ⊆ C, denote by q_{1}, …, q_{s} ∈ Y the points where Y meets the other components of C. Assume that for each Y the following conditions are satisfied:

a. If g(Y) = 1 then s = 1.

b. If g(Y) = 2 then s = 1 and q is not a Weierstrass point.

c. If g(Y) ≥ 3 then (Y, q_{1}, …, q_{s}) is a general s-pointed curve.

Then for p_{1}, … p_{t} ∈ C general points, ρ(l, α^{l}(p_{1}), …, α^{l}(p_{t})) ≥ 0 for any limit linear series on C.

Simple examples involving pointed elliptic curves show that the condition ρ(g, r, d) ≥ ∑_{i=1}^{t} w^{l}(p_{i}) does not guarantee the existence of a linear series l ∈ G_{d}^{r}(C) with prescribed ramification at general points p_{1}, p_{2}, …, p_{t} ∈ C. The appropriate condition in the pointed case can be given in terms of Schubert cycles. Let α = (α_{0}, …, α_{r}) be a Schubert index of type (r, d) and

ℂ^{d+1} = W_{0} ⊃ W_{1} ⊃ … ⊃ W_{d+1} = 0

a decreasing flag of linear subspaces. We consider the Schubert cycle in the Grassmanian,

σ_{α} = {Λ ∈ G(r + 1, d + 1) : dim(Λ ∩ W_{α_{i}+i}) ≥ r + 1 − i, i = 0, …, r}.

For a general t-pointed curve (C, p_{1}, …, p_{t}) of genus g, and α^{1}, …, α^{t} Schubert indices of type (r, d), the necessary and sufficient condition that C has a 𝔤_{d}^{r} with ramification α^{i} at p_{i} is that

σ_{α^{1}} · . . . · σ_{α^{t}} · (σ_{(0,1,... ,1)})^{g} ≠ 0 in H^{∗}(G(r + 1, d + 1), ℤ).  (7)

In the case t = 1 this condition can be made more explicit (cf. [EH3]): a general pointed curve (C, p) of genus g carries a 𝔤_{d}^{r} with ramification sequence (α_{0}, …, α_{r}) at p, if and only if

∑_{i=0}^{r}(α_{i} + g − d + r)_{+} ≤ g,  (8)

where x_{+} = max{x, 0}. One can make the following simple but useful observation:

Proposition 3.3 Let (C, p, q) be a general 2-pointed curve of genus g and (α_{0}, …, α_{r}) a Schubert index of type (r, d). Then C has a 𝔤_{d}^{r} having ramification sequence (α_{0}, …, α_{r}) at p and a cusp at q if and only if

∑_{i=0}^{r}(α_{i} + g + 1 − d + r)_{+} ≤ g + 1.

Proof: The condition for the existence of the 𝔤_{d}^{r} with ramification α at p and a cusp at q is that σ_{α} · (σ_{(0,1,... ,1)})^{g+1} ≠ 0 (cf. (7)). According to the Littlewood-Richardson rule (see [F]), this is equivalent with ∑_{i=0}^{r}(α_{i} + g + 1 − d + r)_{+} ≤ g + 1. □

4 A few consequences of limit linear series

We investigate the Brill-Noether theory of a 2-pointed elliptic curve (see also [EH4]), and we prove that \overline{M}_{g,d}^{r} ∩ Δ_{1} is irreducible for ρ(g, r, d) = −1.

Proposition 4.1 Let (E, p, q) be a two-pointed elliptic curve. Consider the sequences

a : a_{0} < a_{1} < … a_{r} ≤ d, b : b_{0} < b_{1} < … b_{r} ≤ d.

1. For any linear series l = (L, V) ∈ G_{d}^{r}(E) one has that ρ(l, α^{l}(p), α^{l}(q)) ≥ −r. Furthermore, if ρ(l, α^{l}(p), α^{l}(q)) ≤ −1, then p − q ∈ Pic^{0}(E) is a torsion class.

2. Assume that the sequences a and b satisfy the inequalities: d−1 ≤ a_{i} +b_{r−i} ≤ d, i = 0, …, r. Then there exists at most one linear series l ∈ G_{d}^{r}(E) such that a^{l}(p) = a and a^{l}(q) = b. Moreover, there exists exactly one such linear series l = (O_{E}(D), V) with D ∈ E^{(d)}, if and only if for each i = 0, …, r the following is satisfied: if a_{i} + b_{r−i} = d, then D ∼ a_{i} p + b_{r−i} q, and if (a_{i} + 1) p + b_{r−i} q ∼ D, then a_{i+1} = a_{i} + 1.

Proof: In order to prove 1. it is enough to notice that for dimensional reasons there must be sections σ_{i} ∈ V such that div(σ_{i}) ≥ a_{i}^{l}(p) p+a_{r−i}^{l}(q) q, therefore, a_{i}^{l}(p)+a_{r−i}^{l}(q) ≤ d. By adding up all these inequalities, we get that ρ(l, α^{l}(p), α^{l}(q)) ≥ −r. Furthermore, ρ(l, α^{l}(p), α^{l}(q)) ≤ −1 precisely when for at least two values i < j we have equalities a_{i} + b_{r−i} = d, a_{j} + b_{r−j} = d, which means that there are sections σ_{i}, σ_{j} ∈ V such that div(σ_{i}) = a_{i} p + b_{r−i} q, div(σ_{j}) = a_{j} p + b_{r−j} q. By subtracting, we see that p − q ∈ Pic^{0}(E) is torsion. The second part of the Proposition is in fact Prop.5.2 from [EH4]. □

Proposition 4.2 Let g, r, d be such that ρ(g, r, d) = −1. Then the intersection \overline{M}_{g,d}^{r} ∩ Δ_{1} is irreducible.

Proof: Let Y be an irreducible component of \overline{M}_{g,d}^{r}∩Δ_{1}. Either Y ∩IntΔ_{1} ≠ ∅, hence Y = \overline{Y ∩ IntΔ_{1}}, or Y ⊆ Δ_{1} − IntΔ_{1}. The second alternative never occurs. Indeed, if Y ⊆ Δ_{1} − IntΔ_{1}, then since codim (Y, \overline{M_{g}}) = 2, Y must be one of the irreducible components of Δ_{1} − IntΔ_{1}. The components of Δ_{1} − IntΔ_{1} correspond to curves with two nodes. We list these components (see [Ed1]):

• For 1 ≤ j ≤ g − 2, Δ_{1j} is the closure of the locus in \overline{M}_{g} whose general point corresponds to a chain composed of an elliptic curve, a curve of genus g − j − 1, and a curve of genus j.

• The component Δ_{01}, whose general point corresponds to the union of a smooth elliptic curve and an irreducible nodal curve of genus g − 2.

• The component Δ_{0,g−1} whose general point corresponds to the union of a smooth curve of genus g − 1 and an irreducible rational curve.

As the general point of Δ_{1,j}, Δ_{0,1} or Δ_{0,g−1} is a tree-like curve which satisfies the conditions of Prop.3.2 it follows that such a curve satisfies the ‘strong’ Brill-Noether Theorem, hence Δ_{1,j} ⊈ \overline{M}_{g,d}^{r}, Δ_{0,1} ⊈ \overline{M}_{g,d}^{r} and Δ_{0,g−1} ⊈ \overline{M}_{g,d}^{r}, a contradiction. So, we are left with the first possibility: Y = \overline{Y ∩ IntΔ_{1}}. We are going to determine the general point [C] ∈ Y ∩ IntΔ_{1}. Let X = C ∪ E, g(C) = g − 1, E elliptic, E ∩ C = {p} such that X carries a limit 𝔤_{d}^{r}, say l. By the additivity of the Brill-Noether number, we have:

−1 = ρ(g, r, d) ≥ ρ(l, C, p) + ρ(l, E, p).

Since ρ(l, E, p) ≥ 0, it follows that ρ(l, C, p) ≤ −1, so w^{l_{C}}(p) ≥ r. Let us denote by

β : C_{g−1} × C_{1} → IntΔ_{1}

the natural map given by β([C, p], [E, q]) = [X := C ∪ E/p ∼ q]. We claim that if we choose X generically, then α_{0}^{l_{C}}(p) = 0. If not, p is a base point of l_{C} and after removing the base point we get that [C] ∈ M_{g−1,d−1}^{r}. Note that ρ(g − 1, r, d − 1) = −2, so dim M_{g−1,d−1}^{r} = 3g − 8 (cf. [Ed2]). If we denote by π : C_{g−1} → M_{g−1} the morphism which ‘forgets the point’, we get that

dim β(π^{−1}(M_{g−1,d−1}^{r}) × C_{1}) = 3g − 6 < dim Y,

a contradiction. Hence, for the generic [X] ∈ Y we must have α_{0}^{l_{C}}(p) = 0, so a_{r}^{l_{E}}(p) = d. Since an elliptic curve cannot have a meromorphic function with a single pole, it follows that a_{r−1}^{l_{E}}(p) ≤ d − 2 and this implies α^{l_{C}}(p) ≥ (0, 1, …, 1), i.e. l_{C} has a cusp at p. Thus, if we introduce the notation

C_{g−1,d}^{r}(0, 1, …, 1) = {[C, p] ∈ C_{g−1} : G_{d}^{r}(C, (p, (0, 1, …, 1))) ≠ ∅},

then Y ⊆ \overline{β(C_{g−1,d}^{r}(0, 1…. , 1) × C_{1})}. On the other hand, it is known (cf. [EH2]) that C_{g−1,d}^{r}(0, 1, …, 1) is irreducible of dimension 3g − 6 (that is, codimension 1 in C_{g−1}), so we must have Y = \overline{β(C_{g−1,d}^{r}(0, 1, …, 1) × C_{1})}, which not only proves that \overline{M}_{g,d}^{r} ∩ Δ_{1} is irreducible, but also determines the intersection. □

5 The Kodaira dimension of M_{23}

In this section we prove that κ(M_{23}) ≥ 2 and we investigate closely the multicanonical linear systems on \overline{M}_{23}. We now describe the three multicanonical Brill-Noether divisors from Section 2.

5.1 The divisor \overline{M}_{12}^{1}

There is a stratification of M_{23} given by gonality:

M_{2}^{1} ⊆ M_{3}^{1} ⊆ … ⊆ M_{12}^{1} ⊆ M_{23}.

For 2 ≤ d ≤ g/2 + 1 one knows that M_{k}^{1} = M_{g,k}^{1} is an irreducible variety of dimension 2g + 2d − 5. The general point of M_{g,d}^{1} corresponds to a curve having a unique 𝔤_{d}^{1}.

5.2 The divisor \overline{M}_{17}^{2}

The Severi variety V_{d,g} of irreducible plane curves of degree d and geometric genus g, where 0 ≤ g ≤ \binom{d−1}{2}, is an irreducible subscheme of ℙ^{d(d+3)/2} of dimension 3d + g − 1 (cf. [H], [Mod]). Inside V_{d,g} we consider the open dense subset U_{d,g} of irreducible plane curves of degree d having exactly δ = \binom{d−1}{2}−g nodes and no other singularities. There is a global normalization map

m : U_{d,g} → M_{g}, m([Y]) := [Ỹ], Ỹ is the normalization of Y.

When d − 2 ≤ g ≤ \binom{d−1}{2}, d ≥ 5, U_{d,g} has the expected number of moduli, i.e.

dim m(U_{d,g}) = min(3g − 3, 3g − 3 + ρ(g, 2, d)).

In our case we can summarize this as follows:

Proposition 5.1 There is exactly one component of G_{17}^{2} mapping dominantly to M_{17}^{2}. The general element (C, l) ∈ G_{17}^{2} corresponds to a curve C of genus 23, together with a 𝔤_{17}^{2} which provides a plane model for C of degree 17 with 97 nodes.

5.3 The divisor \overline{M}_{20}^{3}

Here we combine the result of Eisenbud and Harris (see [EH2]) about the uniqueness of divisorial components of G_{d}^{r} when ρ(g, r, d) = −1, with Sernesi’s (see [Se2]) which asserts the existence of components of the Hilbert scheme H_{d,g} parametrizing curves in ℙ^{3} of degree d and genus g with the expected number of moduli, for d − 3 ≤ g ≤ 3d − 18, d ≥ 9.

Proposition 5.2 There is exactly one component of G_{20}^{3} mapping dominantly to M_{20}^{3}. The general point of this component corresponds to a pair (C, l) where C is a curve of genus 23 and l is a very ample 𝔤_{20}^{3}.

We are going to prove that the Brill-Noether divisors \overline{M}_{12}^{1}, \overline{M}_{17}^{2} and \overline{M}_{20}^{3} are mutually distinct.

Theorem 2 There exists a smooth curve of genus 23 having a 𝔤_{17}^{2}, but no 𝔤_{20}^{3}’s. Equivalently, one has supp(M_{17}^{2}) ⊈ supp(M_{20}^{3}).

Proof: It suffices to construct a reducible curve X of compact type of genus 23, which has a smoothable limit 𝔤_{17}^{2}, but no limit 𝔤_{20}^{3}. If [C] ∈ M_{23} is a nearby smoothing of X which preserves the 𝔤_{17}^{2}, then [C] ∈ M_{17}^{2} − M_{20}^{3}. Let us consider the following curve:

[figure: C_{1}; p_{1}; p_{2}; C_{2}; E]
X := C_{1} ∪ C_{2} ∪ E,

where (C_{1}, p_{1}) and (C_{2}, p_{2}) are general pointed curves of genus 11, E is an elliptic curve, and p_{1} − p_{2} is a primitive 9-torsion point in Pic^{0}(E) Step 1) There is no limit 𝔤_{20}^{3} on X. Assume that l is a limit 𝔤_{20}^{3} on X. By the additivity of the Brill-Noether number,

−1 ≥ ρ(l_{C_{1}}, p_{1}) + ρ(l_{C_{2}}, p_{2}) + ρ(l_{E}, p_{1}, p_{2}).

Since (C_{i}, p_{i}) are general points in C_{11}, it follows from Prop.3.2 that ρ(l_{C_{i}}, p_{i}) ≥ 0, hence ρ(l_{E}, p_{1}, p_{2}) ≤ −1. On the other hand ρ(l_{E}, p_{1}, p_{2}) ≥ −3 from Prop.4.1.

Denote by (a_{0}, a_{1}, a_{2}, a_{3}) the vanishing sequence of l_{E} at p_{1}, and by (b_{0}, b_{1}, b_{2}, b_{3}) that of l_{E} at p_{2}. The condition (8) for a general pointed curve [(C_{i}, p_{i})] ∈ C_{11} to possess a 𝔤_{20}^{3} with prescribed ramification at the point p_{i} and the compatibility conditions between l_{C_{i}} and l_{E} at p_{i} give that:

(14 − a_{3})_{+} + (13 − a_{2})_{+} + (12 − a_{1})_{+} + (11 − a_{0})_{+} ≤ 11,  (9)

and

(14 − b_{3})_{+} + (13 − b_{2})_{+} + (12 − b_{1})_{+} + (11 − b_{0})_{+} ≤ 11.  (10)

1st case: ρ(l_{E}, p_{1}, p_{2}) = −3. Then a_{i} + b_{3−i} = 20, for i = 0, …, 3 and it immediately follows that 20(p_{1} − p_{2}) ∼ 0 in Pic^{0}(E), a contradiction. 2nd case: ρ(l_{E}, p_{1}, p_{2}) = −2. We have two distinct possibilities here: i) a_{0} + b_{3} = 20, a_{1} + b_{2} = 20, a_{2} + b_{1} = 20, a_{3} + b_{0} = 19. Then it follows that a^{l_{E}}(p_{1}) = (0, 9, 18, 19) and a^{l_{E}}(p_{2}) = (0, 2, 11, 20), while according to (9), a_{3} ≤ 15, (because ρ(l_{C_{1}}, p_{1}) ≤ 1), a contradiction. ii) a_{0} + b_{3} = 20, a_{1} + b_{2} = 20, a_{2} + b_{1} = 19, a_{3} + b_{0} = 20. Again, it follows that a_{3} = a_{0} + 18 ≥ 15, a contradiction. 3rd case: ρ(l_{E}, p_{1}, p_{2}) = −1. Then ρ(l_{C_{i}}, p_{i}) = 0 and l is a refined limit 𝔤_{20}^{3}. From (9) and (10) we must have: a^{l_{E}}(p_{i}) ≤ (11, 12, 13, 14), i = 1, 2. There are four possibilities: i) a_{0} + b_{3} = a_{1} + b_{2} = 20, a_{2} + b_{1} = a_{3} + b_{0} = 19. Then a_{1} = a_{0} + 9 ≤ 12, so b_{3} = 20 − a_{0} ≥ 17, a contradiction. ii) a_{0} + b_{3} = a_{2} + b_{1} = 20, a_{2} + b_{1} = a_{3} + b_{0} = 19. Then b_{3} = 20 − a_{0} ≤ 14, so a_{2} = a_{0} + 9 ≥ 15, a contradiction. iii) a_{0} + b_{3} = a_{3} + b_{0} = 20, a_{1} +b_{2} = a_{2} +b_{1} = 19. Then b_{3} = 19−a_{0} ≤ 14, so a_{3} ≥ a_{0} +9 ≥ 15, a contradiction. iv) a_{0} + b_{3} = a_{3} + b_{0} = 19, a_{1} + b_{2} = a_{2} + b_{1} = 20. Then b_{3} = 19 − a_{0} ≤ 14, so a_{2} ≥ a_{1} + 9 ≥ 15, a contradiction again. We conclude that X has no limit 𝔤_{20}^{3}.

Step 2) There exists a smoothable limit 𝔤_{17}^{2} on X, hence [X] ∈ \overline{M}_{17}^{2}. We construct a limit linear series l of type 𝔤_{17}^{2} on X, aspect by aspect: on C_{i} take l_{C_{i}} ∈ G_{17}^{2}(C_{i}) such that a^{l_{C_{i}}}(p_{i}) = (4, 9, 13). Note that in this case ∑_{j=0}^{r}(α_{j} + g − d + r)_{+} = g, so (8) ensures the existence of such a 𝔤_{17}^{2}. On E we take l_{E} = |V_{E}|, where |V_{E}| ⊆ |4p_{1} + 13p_{2}| = |4p_{2} + 13p_{1}| is a 𝔤_{17}^{2} with vanishing sequence (4, 8, 13) at p_{i}. Prop.4.1 ensures the existence of such a linear series. In this way l is a refined limit 𝔤_{17}^{2} on X with ρ(l_{C_{i}}, p_{i}) = 0, ρ(l_{E}, p_{1}, p_{2}) = −1. We prove that l is dimensionally proper. Let π_{i} : C_{i} → Δ_{i}, p̃_{i} : Δ_{i} → C_{i}, be the versal deformation of [(C_{i}, p_{i})] ∈ C_{11}, and σ_{i} : G_{17}^{2}(C_{i}/Δ_{i}, (p̃_{i}, (4, 8, 11))) → Δ_{i} the projection.

Since being general is an open condition, we have that σ_{i} is surjective and dim σ_{i}^{−1}(t) = ρ(l_{C_{i}}, p_{i}) = 0, for each t ∈ Δ_{i}, therefore

dim G_{17}^{2}(C_{i}/Δ_{i}, (p̃_{i}, (4, 8, 11))) = dim Δ_{i} + ρ(l_{C_{i}}, p_{i}) = 31.

Next, let π : C → Δ, p̃_{1}, p̃_{2} : Δ → C be the versal deformation of (E, p_{1}, p_{2}). We prove that

dim G_{17}^{2}(C/Δ, (p̃_{i}, (4, 7, 11))) = dim Δ + ρ(l_{E}, p_{1}, p_{2}) = 1.

This follows from Prop.4.1, since a 2-pointed elliptic curve (E_{t}, p̃_{1}(t), p̃_{2}(t)) has at most one 𝔤_{17}^{2} with ramification (4, 7, 11) at both p̃_{1}(t) and p̃_{2}(t), and exactly one when 9(p̃_{1}(t) − p̃_{2}(t)) ∼ 0. Hence ImG_{17}^{2}(C/Δ, (p̃_{i}, (4, 7, 11))) = {t ∈ Δ : 9(p̃_{1}(t) − p̃_{2}(t)) ∼ 0 in Pic^{0}(E_{t})}, which is a divisor on Δ, so the claim follows and l is a dimensionally proper 𝔤_{17}^{2}. □

A slight variation of the previous argument gives us:

Proposition 5.3 We have supp(\overline{M}_{17}^{2} ∩ Δ_{1}) ≠ supp(\overline{M}_{20}^{3} ∩ Δ_{1}).

Proof: We construct a curve [Y] ∈ Δ_{1} ⊆ \overline{M}_{23} which has a smoothable limit 𝔤_{17}^{2} but no limit 𝔤_{20}^{3}. Let us consider the following curve:

[figure: x; p_{1}; p_{2}; E_{1}; C_{1}; C_{2}; E]
Y := C_{1} ∪ C_{2} ∪ E_{1} ∪ E,

where (C_{2}, p_{2}) is a general point of C_{11}, (C_{1}, p_{1}, x) is a general 2-pointed curve of genus 10, (E_{1}, x) is general in C_{1}, E is an elliptic curve, and p_{1} −p_{2} ∈ Pic^{0}(E) is a primitive 9torsion. In order to prove that Y has no limit 𝔤_{20}^{3}, one just has to take into account that according to Prop.3.3, the condition for a general 1-pointed curve (C, z) of genus g, to have a 𝔤_{d}^{r} with ramification α at z is the same with the condition for a general 2-pointed curve (D, x, y) of genus g − 1 to have a 𝔤_{d}^{r} with ramification α at x and a cusp at y. Therefore we can repeat what we did in the proof of Theorem 2. Next, we construct l, a smoothable limit 𝔤_{17}^{2} on Y: take l_{C_{2}} ∈ G_{17}^{2}(C_{2}, (p_{2}, (4, 8, 11))), l_{E} = |V_{E}| ⊆ |4p_{1} + 13p_{2}|, with α^{l_{E}}(p_{i}) = (4, 7, 11), on E_{1} take l_{E_{1}} = 14x + |3x|, and finally on C_{1} take l_{C_{1}} such that α^{l_{C_{1}}}(p_{1}) = (4, 8, 11), α^{l_{C_{1}}}(x) = (0, 0, 1). Prop.3.3 ensures the existence of l_{C_{1}}. Clearly, l is a refined limit 𝔤_{17}^{2} and the proof that it is smoothable is all but identical to the one in the last part of Theorem 2. □

The other cases are settled by the following:

Theorem 3 There exists a smooth curve of genus 23 having a 𝔤_{12}^{1} but having no 𝔤_{17}^{2} nor 𝔤_{20}^{3}. Equivalently, supp(M_{12}^{1}) ⊈ supp(M_{17}^{2}) and supp(M_{12}^{1}) ⊈ supp(M_{20}^{3}).

Proof: We take the curve considered in [EH3]:

[figure: C_{1}; p_{1}; p_{2}; C_{2}; E]
Y := C_{1} ∪ C_{2} ∪ E,

where (C_{i}, p_{i}) are general points of C_{11}, E is elliptic and p_{1} −p_{2} ∈ Pic^{0}(E) is a primitive 12-torsion. Clearly Y has a (smoothable) limit 𝔤_{12}^{1}: on C_{i} take the pencil |12p_{i}|, while on E take the pencil spanned by 12p_{1} and 12p_{2}. It is proved in [EH3] that Y has no limit 𝔤_{17}^{2}’s and similarly one can prove that Y has no limit 𝔤_{20}^{3}’s either. We omit the details.□

Now we are going to prove that equation (2)

supp(M_{12}^{1}) ∩ supp(M_{17}^{2}) = supp(M_{17}^{2}) ∩ supp(M_{20}^{3}) = supp(M_{20}^{3}) ∩ supp(M_{12}^{1})

is impossible, and as explained before, this will imply that κ(M_{23}) ≥ 2. The main step in this direction is the following:

Proposition 5.4 There exists a stable curve of compact type of genus 23 which has a smoothable limit 𝔤_{20}^{3}, a smoothable limit 𝔤_{15}^{2} (therefore also a 𝔤_{17}^{2}), but has generic gonality, that is, it does not have any limit 𝔤_{12}^{1}.

Proof We shall consider the following stable curve X of genus 23:

… Γ E_{1} E_{2}  E_{8} p_{1} p_{2}  p_{8}
X := Γ ∪ E_{1} ∪ … ∪ E_{8},

where the E_{i}’s are elliptic curves, Γ ⊆ ℙ^{2} is a general smooth plane septic and the points of attachment {p_{i}} = Γ ∪ E_{i} are general points of Γ.

Step 1) There is no limit 𝔤_{12}^{1} on X. Assume that l is a limit 𝔤_{12}^{1} on X. Since the elliptic curves E_{i} cannot have meromorphic functions with a single pole, we have that a^{l_{E_{i}}}(p_{i}) ≤ (10, 12), hence α^{l_{Γ}}(p_{i}) ≥ (0, 1), that is, l_{Γ} has a cusp at p_{i} for i = 1, …, 8. We now prove that Γ has no 𝔤_{12}^{1}’s with cusps at the points p_{i}.

First, we notice that dim G_{12}^{1}(Γ) = ρ(15, 1, 12) = 7. Indeed, if we assume that dim G_{12}^{1}(Γ) ≥ 8, by applying Keem’s Theorem (cf. [ACGH], p.200) we would get that Γ possesses a 𝔤_{4}^{1}, which is impossible since gon(Γ) = 6. (In general, if Y ⊆ ℙ^{2} is a smooth plane curve, deg(Y) = d, then gon (Y) = d − 1, and the 𝔤_{d−1}^{1} computing the gonality is cut out by the lines passing through a point p ∈ Y, see [ACGH].) Next, we define the variety

Σ = {(l, q_{1}, …, q_{8}) ∈ G_{12}^{1}(Γ) × Γ^{8} : α^{l}(q_{i}) ≥ (0, 1), i = 1, …, 8}

and denote by π_{1} : Σ → G_{12}^{1}(Γ) and π_{2} : Σ → Γ^{8} the two projections. For any l ∈ G_{12}^{1}(Γ), the fibre π_{1}^{−1}(l) is finite, hence dim Σ = dim G_{12}^{1}(Γ) = 7, which shows that π_{2} cannot be surjective and this proves our claim. Step 2) There exists a smoothable limit 𝔤_{15}^{2} on X, hence [X] ∈ \overline{M}_{15}^{2}. We construct l, a limit 𝔤_{15}^{2} on X as follows: on Γ there is a (unique) 𝔤_{7}^{2}, and we consider l_{Γ} = 𝔤_{7}^{2}(p_{1} + ⋯ + p_{8}), i.e. the Γ− aspect l_{Γ} is obtained from the 𝔤_{7}^{2} by adding the base points p_{1}, …, p_{8}. Clearly a^{l_{Γ}}(p_{i}) = (1, 2, 3) for each i. On E_{i} we take l_{E_{i}} = 𝔤_{3}^{2}(12p_{i}) for i = 1, …, 8, where 𝔤_{3}^{2} is a complete linear series of the form |2p_{i} + x_{i}|, with x_{i} ∈ E_{i} − {p_{i}}. Furthermore, a^{l_{E_{i}}}(p_{i}) = (12, 13, 14), so l = {l_{Γ}, l_{E_{i}}} is a refined limit 𝔤_{15}^{2} on X. One sees that ρ(l_{E_{i}}, α^{l_{E_{i}}}(p_{i})) = 1 for all i, ρ(l_{Γ}, α^{l_{Γ}}(p_{1}), …, α^{l_{Γ}}(p_{8})) = −15, and ρ(l) = −7. We now prove that l is dimensionally proper.

Let π_{i} : C_{i} → Δ_{i}, p̃_{i} : Δ_{i} → C_{i} be the versal deformation space of (E_{i}, p_{i}), for i = 1, …, 8. There is an obvious isomorphism over Δ_{i}

G_{15}^{2}(C_{i}/Δ_{i}, (p̃_{i}, (12, 12, 12))) ≃ G_{3}^{2}(C_{i}/Δ_{i}, (p̃_{i}, 0)).

If σ_{i} : G_{3}^{2}(C_{i}/Δ_{i}, (p̃_{i}, 0)) → Δ_{i} is the natural projection, then for each t ∈ Δ_{i}, the fibre σ_{i}^{−1}(t) is isomorphic to π_{i}^{−1}(t), the isomorphism being given by

π_{i}^{−1}(t) ∋ q ↦ |2p̃_{i}(t) + q| ∈ G_{3}^{2}(π_{i}^{−1}(t)).

Thus, G_{3}^{2}(C_{i}/Δ_{i}, (p̃_{i}, 0)) is a smooth irreducible surface, which shows that l is dimensionally proper w.r.t. E_{i}. Next, let us consider π : X → Δ, p̃_{1}, …, p̃_{8} : Δ → X, the versal deformation of (Γ, p_{1}, …, p_{8}). We have to prove that

dim G_{15}^{2}(X /Δ, (p̃_{i}, (1, 1, 1))) = dim Δ + ρ(l_{Γ}, α^{l_{Γ}}(p_{i})) = 35.

There is an isomorphism over Δ,

G_{15}^{2}(X /Δ, (p̃_{i}, (1, 1, 1))) ≃ G_{7}^{2}(X /Δ, (p̃_{i}, 0)).

If π_{0} : C → M is the versal deformation space of Γ, then we denote by G_{7}^{2} → M the scheme parametrizing 𝔤_{7}^{2}’s on curves of genus 15 ‘nearby’ Γ (See Section 3 for this notation). Clearly G_{7}^{2}(X /Δ, (p̃_{i}, 0)) ≃ G_{7}^{2} ×_{M} Δ, so it suffices to prove that G_{7}^{2} has the expected dimension at the point (Γ, 𝔤_{7}^{2}). For this we use Prop.3.1. We have that N_{Γ/ℙ^{2}} = O_{Γ}(7), K_{Γ} = O_{Γ}(4), hence

H^{1}(Γ, N_{Γ/ℙ^{2}}) ≃ H^{0}(Γ, O_{Γ}(−3))^{∨} = 0,

so l is dimensionally proper w.r.t. Γ as well. We conclude that l is smoothable.

Step 3) There exists a smoothable limit 𝔤_{20}^{3} on X, that is [X] ∈ \overline{M}_{20}^{3}. First we notice that there is an isomorphism Γ ⟶^{∼} G_{6}^{1}(Γ), given by

Γ ∋ p ↦ |𝔤_{7}^{2} − p| ∈ G_{6}^{1}(Γ).

Consequently, there is a 2-dimensional family of 𝔤_{12}^{3}’s on Γ, of the form 𝔤_{12}^{3} = 𝔤_{6}^{1} + 𝔥_{6}^{1} = |2𝔤_{7}^{2} − p − q|, where p, q ∈ Γ. Pick l_{0} = l′_{0} + l′′_{0}, with l′_{0}, l′′_{0} ∈ G_{6}^{1}(Γ), a general 𝔤_{12}^{3} of this type. We construct l, a limit 𝔤_{20}^{3} on X, as follows: the Γ-aspect is given by l_{Γ} = l_{0}(p_{1} + ⋯ p_{8}), and because of the generality of the chosen l_{0} we have that ρ(l_{Γ}, α^{l_{Γ}}(p_{1}), …, α^{l_{Γ}}(p_{8})) = −9. The E_{i}-aspect is given by l_{E_{i}} = 𝔤_{4}^{3}(16p_{i}), where 𝔤_{4}^{3} = |3p_{i} + x_{i}|, with x_{i} ∈ E_{i} − {p_{i}}, for i = 1, …, 8. It is clear that ρ(l_{E_{i}}, α^{l_{E_{i}}}(p_{i})) = 1 and that l′ = {l_{Γ}, l_{E_{i}}} is a refined limit 𝔤_{20}^{3} on X.

In order to prove that l′ is dimensionally proper, we first notice that l′ is dimensionally proper w.r.t. the elliptic tails E_{i}. We now prove that l′ is dimensionally proper w.r.t. Γ. As in the previous step, we consider π : X → Δ, p̃_{1}, …, p̃_{8} : Δ → X , the versal deformation of (Γ, p_{1}, …, p_{8}) and π_{0} : C → M, the versal deformation space of Γ. There is an isomorphism over Δ

G_{20}^{3}(X /Δ, (p̃_{1}, α^{l_{Γ}}(p_{1}), …, (p̃_{8}, α^{l_{Γ}}(p_{8}))) ≃ G_{12}^{3}(C/M) ×_{M} Δ.

It suffices to prove that G_{12}^{3} = G_{12}^{3}(C/M) has a component of the expected dimension passing through (Γ, l_{0}). In this way, the genus 23 problem is turned into a deformation theoretic problem in genus 15. Denote as usual by σ : G_{12}^{3} → M the natural projection. According to Prop.3.1, it will be enough to exhibit an element (C, l) ∈ G_{20}^{3}, sitting in the same component as (Γ, l_{0}), such that the linear system l is base point free and simple, and if ϕ_{1} : C → ℙ^{3} is the map induced by l, then H^{1}(C, N_{ϕ_{1}}) = 0. Certainly we cannot take C to be a smooth plane septic because in this case H^{1}(C, N_{ϕ_{1}}) ≠ 0, as one can easily see. Instead, we consider the 6-gonal locus in a neighbourhood of the point [Γ] ∈ M_{15}, or equivalently, the 6-gonal locus in M, the versal deformation space of Γ. One has the projection G_{6}^{1} → M and the scheme G_{6}^{1} is smooth (and irreducible) of dimension 37(= 2g + 2d − 5; g = 15, d = 6). We denote by

μ : G_{6}^{1} ×_{M} G_{6}^{1} → M, μ([C, l, l′]) = [C].

There is a stratification of M given by the number of pencils: for i ≥ 0 we define,

M(i)^{0} := {[C] ∈ M : C possesses i mutually independent, base-point-free 𝔤_{6}^{1}’s },

and M(i) := \overline{M(i)^{0}}. The strata M(i)^{0} are constructible subsets of M, the first stratum M(1) = Im (G_{6}^{1}) is just the 6-gonal locus; the stratum M(2) is irreducible and dim M(2) = g + 4d − 7 = 32 (cf. [AC1]). We denote by M_{sept} := \overline{m(U_{7,15}) ∩ M}, the closure of the locus of smooth plane septics in M, and by M_{oct} := \overline{m(U_{8,15}) ∩ M}, the closure of the locus of curves which are normalizations of plane octics with 6 nodes. Since the Severi varieties U_{7,15} and U_{8,15} are irreducible, so are the loci M_{sept} and M_{oct}. Furthermore dim M_{sept} = 27 and dim M_{oct} = 30. We prove that M_{sept} ⊆ M_{oct}. Indeed, let us pick Y ⊆ ℙ^{2} a smooth plane septic, and L ⊆ ℙ^{2} a general line, L · Y = p_{1} + ⋯ + p_{7}. Denote Z := C ∪ L, deg (Z) = 8, p_{a}(Z) = 21. We consider the node p_{7} unassigned, while p_{1}, … p_{6} are assigned. By using [Ta] Theorem 2.13, there exists a flat family of plane curves π : Z → B and a point 0 ∈ B, such that Z_{0} = π^{−1}(0) = Z, while for 0 ≠ b ∈ B, the fibre Z_{b} is an irreducible octic with nodes p_{1}(b), … p_{6}(b), and such that p_{i}(b) → p_{i}, when b → 0, for i = 1, …, 6. If Z′ → B is the family resulting by normalizing the surface Z, and η : Z′′ → B is the stable family associated to the semistable family Z′ → B, then we get that η^{−1}(0) = Y, while η^{−1}(b) is the normalization of Z_{b} for b ≠ 0. This proves our contention.

Since M_{oct} is irreducible there is a component A of G_{6}^{1} ×_{M} G_{6}^{1}, such that μ(A) ⊇ M_{oct}. The general point of A corresponds to a curve C and two base-point-free pencils l′, l′′ ∈ G_{6}^{1}(C) such that if f′ : C → ℙ^{1} and f′′ : C → ℙ^{1} are the corresponding morphisms, then

ϕ = (f′, f′′) : C → ℙ^{1} × ℙ^{1}

is birational. Since [Γ] ∈ μ(A) we can assume that [Γ, l′_{0}, l′′_{0}] ∈ A. As a matter of fact, we can start the construction of a limit 𝔤_{20}^{3} on the genus 23 curve X, by taking any pair of base-point free pencils (l′_{0}, l′′_{0}) ∈ G_{6}^{1}(Γ) × G_{6}^{1}(Γ) such that dim|l′_{0} + l′′_{0}| = 3, the argument does not change.

We denote by η : A → G_{12}^{3} the map given by η(C, l′, l′′) := (C, l′ + l′′). The fact that η maps to G_{12}^{3} follows from the base-point-free-pencil-trick.

We are going to show that given a general point [C] ∈ M_{oct} and (C, l, l′) ∈ μ^{−1}([C]), the condition H^{1}(C, N_{ϕ_{1}}) = 0 is satisfied, hence G_{12}^{3} is smooth of the expected dimension at the point (C, l + l′). This will prove the existence of a component of G_{12}^{3} passing through (Γ, l_{0}) and having the expected dimension. We take \overline{C} ⊆ ℙ^{2}, a general point of U_{8,15}, with nodes p_{1}, …, p_{6} ∈ ℙ^{2} in general position. Theorem 3.2 from [AC1] ensures that there exists a plane octic having 6 prescribed nodes in general position. Let ν : C → \overline{C} be the normalization map, ν^{−1}(p_{i}) = q′_{i} + q′′_{i} for i = 1, …, 6. Choose two nodes, say p_{1} and p_{2}, and denote by 𝔤_{6}^{1} = |H − q′_{1} − q′′_{1}| and 𝔥_{6}^{1} = |H − q′_{2} − q′′_{2}|, the linear series obtained by projecting \overline{C} from p_{1} and p_{2} respectively. Here H ∈ |ν^{∗}O_{ℙ^{2}}(1)| is an arbitrary line section of C. The morphism induced by (𝔤_{6}^{1}, 𝔥_{6}^{1}) is denoted by ϕ : C → ℙ^{1} × ℙ^{1}, and ϕ_{1} = s ∘ ϕ : C → ℙ^{3}, with s : ℙ^{1} × ℙ^{1} → ℙ^{3} the Segre embedding. There is an exact sequence over C

0 ⟶ N_{ϕ} ⟶ N_{ϕ_{1}} ⟶ ϕ^{∗}N_{ℙ^{1}×ℙ^{1}/ℙ^{3}} ⟶ 0.  (11)

We can argue as in [AC2] p.473, that for a general (C, 𝔤_{6}^{1}, 𝔥_{6}^{1}) with [C] ∈ M_{oct}, we have h^{1}(C, N_{ϕ}) = 0. Indeed, let us denote by A_{0} the open set of A corresponding to points (X, l, l′) such that χ : X → ℙ^{1} × ℙ^{1}, the morphism associated to the pair of pencils (l, l′) is birational, and by U ⊆ A_{0} the variety of those points (X, l, l′) ∈ A_{0} such that H^{1}(X, N_{χ}) ≠ 0. Define

V := {x = (X, l, l′, F, F′) : (X, l, l′) ∈ U, F is a frame for l, F′ is a frame for l′}.

We may assume that for a generic x ∈ U, the corresponding pencils l and l′ are basepoint-free. Suppose that U has a component of dimension α. For any x ∈ V,

T_{x}(V) ⊆ H^{0}(X, N_{χ}), and dim T_{x}(V) ≥ α + 2 dim P GL(2) = α + 6.

If K_{χ} is the cuspidal sheaf of χ and N′_{χ} = N_{χ}/K_{χ}, then according to [AC1] Lemma 1.4, for a general point x ∈ V one has that,

T_{x}(V) ∩ H^{0}(X, K_{χ}) = 0,

from which it follows that α ≤ g − 6. If not, one would have that h^{0}(X, N′_{χ}) ≥ g + 1, and therefore by Clifford’s Theorem, h^{1}(X, N_{χ}) = h^{1}(X, N′_{χ}) = 0, which contradicts the definition of U. Since clearly dim M_{oct} > g −6, we can assume that h^{1}(C, N_{ϕ}) = 0, for the general [C] ∈ M_{oct}. Therefore, by taking cohomology in (11), we get that

H^{1}(C, N_{ϕ_{1}}) = H^{1}(C, O_{C}(2)),

where O_{C}(1) = ϕ_{1}^{∗}O_{ℙ^{3}}(1). By Serre duality,

H^{1}(C, O_{C}(2)) = 0 ⟺ |K_{C} − 2𝔤_{6}^{1} − 2𝔥_{6}^{1}| = ∅.  (12)

Since K_{C} = 5H − ∑_{i=1}^{6}(q′_{i} + q′′_{i}), equation (12) becomes

|H + q′_{1} + q′′_{1} + q′_{2} + q′′_{2} − ∑_{i=3}^{6}(q′_{i} + q′′_{i})| = ∅.  (13)

If L = \overline{p_{1}p_{2}} ⊆ ℙ^{2}, we can write ν^{∗}(L) = q′_{1} + q′′_{1} + q′_{2} + q′′_{2} + x + y + z + t, and (13) is rewritten as

|2H − x − y − z − t − ∑_{i=3}^{6}(q′_{i} + q′′_{i})| = ∅.

So, one has to show that there are no conics passing through the nodes p_{3}, p_{4}, p_{5} and p_{6} and also through the points in L · \overline{C} − 2p_{1} − 2p_{2}. Because [\overline{C}] ∈ U_{8,15} is general we may assume that x, y, z and t are distinct, smooth points of \overline{C}. Indeed, if the divisor x + y + z + t on \overline{C} does not consist of distinct points, or one of its points is a node, we obtain that \overline{C} has intersection number 8 with the line L at 5 points or less. But according to [DH], the locus in the Severi variety

{[X] ∈ U_{d,g} : X has total intersection number m + 3 with a line at m points }

is a divisor on U_{d,g}, so we may assume that [\overline{C}] lies outside this divisor. Now, if x, y, z and t are distinct and smooth points of \overline{C}, a conic satisfying (13) would necessarily be a degenerate one, and one gets a contradiction with the assumption that the nodes p_{1}, …, p_{6} of \overline{C} are in general position. □ Remark: We have a nice geometric characterization of some of the strata M_{i}. First, by using Zariski’s Main Theorem for the birational projection G_{6}^{1} → M(1), one sees that [C] ∈ M(1)_{sing} if and only if either [C] ∈ M(2)^{0}, or C possesses a 𝔤_{6}^{1} such that dim |2𝔤_{6}^{1}| ≥ 3. In the latter case, the 𝔤_{6}^{1} is a specialization of 2 different 𝔤_{6}^{1}’s in some family of curves, hence M(2) = M(1)_{sing} (cf [Co2]). As a matter of fact, Coppens has proved that for 4 ≤ k ≤ [(g +1)/2] and 8 ≤ g ≤ (k −1)^{2}, there exists a k-gonal curve of genus g carrying exactly 2 linear series 𝔤_{k}^{1}, so the general point of M(2) corresponds to a curve C of genus 15, having exactly 2 base-point-free 𝔤_{6}^{1}’s. Furthermore, using Coppens’ classification of curves having many pencils computing the gonality (see [Co1]), we have that M(6) = M_{oct}, and M(i) = M_{sept}, for each i ≥ 7.

Now we are in a position to complete the proof of Theorem 1: Proof of Theorem 1 According to (2), it will suffice to prove that there exists a smooth curve [Y] ∈ M_{23} which carries a 𝔤_{20}^{3}, a 𝔤_{17}^{2} but has no 𝔤_{12}^{1}’s. In the proof of Prop.5.4 we constructed a stable curve of compact type [X] ∈ \overline{M}_{23} such that [X] ∈ \overline{M}_{17}^{2} ∩ \overline{M}_{20}^{3}, but [X] ∉ \overline{M}_{12}^{1}. If we prove that [X] ∈ \overline{M_{17}^{2} ∩ M_{20}^{3}}, that is, there are smoothings of X which preserve both the 𝔤_{17}^{2} and the 𝔤_{20}^{3}, we are done. One can write \overline{M}_{17}^{2} ∩ \overline{M}_{20}^{3} = Y_{1} ∪ … ∪ Y_{s}, where Y_{i} are irreducible codimension 2 subvarieties of \overline{M}_{23}. Assume that [X] ∈ Y_{1}. If Y_{1} ∩ M_{23} ≠ ∅, then [X] ∈ Y_{1} = \overline{Y_{1} ∩ M_{23}} ⊆ \overline{M_{17}^{2} ∩ M_{20}^{3}}, and the conclusion follows. So we may assume that Y_{1} ⊆ \overline{M}_{23} − M_{23}. Because [X] ∈ Δ_{1} − ⋃_{j≠1} Δ_{j}, we must have Y ⊆ Δ_{1}. It follows that \overline{M}_{17}^{2} ∩ Δ_{1} and \overline{M}_{20}^{3} ∩ Δ_{1} have Y_{1} as a common component. According to Prop.4.2, both intersections \overline{M}_{17}^{2} ∩ Δ_{1} and \overline{M}_{20}^{3} ∩ Δ_{1} are irreducible, hence \overline{M}_{17}^{2} ∩ Δ_{1} = \overline{M}_{20}^{3} ∩ Δ_{1} = Y_{1}, which contradicts Prop.5.3. Theorem 1 now follows. □

6 The slope conjecture and M_{23}

In this final section we explain how the slope conjecture in the context of M_{23} implies that κ(M_{23}) = 2, and then we present evidence for this.

The slope of \overline{M}_{g} is defined as s_{g} := inf \{a ∈ ℝ_{>0} : |aλ − δ| ≠ ∅\}, where δ = δ_{0} + δ_{1} + ⋯ + δ_{[g/2]}, λ ∈ Pic(\overline{M}_{g}) ⊗ ℝ. Since λ is big, it follows that s_{g} < ∞. If 𝔼 is the cone of effective divisors in Div(\overline{M}_{g}) ⊗ ℝ, we define the slope function s : 𝔼 → ℝ by the formula

s_{D} := inf {a/b : a, b > 0 such that ∃c_{i} ≥ 0 with [D] = aλ − bδ − ∑_{i=0}^{[g/2]} c_{i}δ_{i}},

for an effective divisor D on \overline{M}_{g}. Clearly s_{g} ≤ s_{D} for any D ∈ 𝔼. When g + 1 is composite, we obtain the estimate s_{g} ≤ 6 + 12/(g + 1) by using the Brill-Noether divisors \overline{M}_{g,d}^{r}, with ρ(g, r, d) = −1.

Conjecture 1 ([HMo]) We have that s_{g} ≥ 6+12/(g+1) for each g ≥ 3, with equality when g + 1 is composite.

Harris and Morrison also stated (in a somewhat vague form) that for composite g + 1, the Brill-Noether divisors not only minimize the slope among all effective divisors, but they also single out those irreducible D ∈ 𝔼 with s_{D} = s_{g}.

The slope conjecture has been proved for 3 ≤ g ≤ 11, g ≠ 10 (cf. [HMo], [CR3,4], [Tan]). A strong form of the conjecture holds for g = 3 and g = 5: on \overline{M}_{3} the only irreducible divisor of slope s_{3} = 9 is the hyperelliptic divisor, while on \overline{M}_{5} the only irreducible divisor of slope s_{5} = 8 is the trigonal divisor (cf. [HMo]). Conjecture 1 would imply that κ(M_{g}) = −∞ for all g ≤ 22. For g = 23, we rewrite (1) as

nK_{\overline{M}_{23}} = \frac{n}{c_{23,r,d}} [\overline{M}_{g,d}^{r}] + 8n δ_{1} + ∑_{i=2}^{11} \frac{(i(23 − i) − 4)}{2}n δ_{i} (n ≥ 1),  (14)

(see Section 2 for the coefficients c_{g,r,d}). As Harris and Morrison suggest, we can ask the question whether the class of any D ∈ 𝔼 with s_{D} = s_{g} is (modulo a sum of boundary components Δ_{i}) proportional to [\overline{M}_{23,d}^{r}], and whether the sections defining (multiples of) \overline{M}_{23,d}^{r} form a maximal algebraically independent subset of the canonical ring R(\overline{M}_{23}). If so, it would mean that the boundary divisor 8nδ_{1}+(1/2) ∑_{i=2}^{11} n(i(23− i) − 4)δ_{i} is a fixed part of |nK_{\overline{M}_{23}}|. Moreover, using our independence result for the three Brill-Noether divisors, it would follow that h^{0}(\overline{M}_{23}, nK_{\overline{M}_{23}}) grows quadratically in n, for n sufficiently high and sufficiently divisible, hence κ(M_{23}) = 2. We would also have that Σ ∩ M_{23} = M_{12}^{1} ∩ M_{17}^{2} ∩ M_{20}^{3}, with Σ the common base locus of all the linear systems |nK_{\overline{M}_{23}}|.

Evidence for these facts is of various sorts: first, one knows (cf. [Tan], [CR3]) that |nK_{\overline{M}_{23}}| has a large fixed part in the boundary: for each n ≥ 1, every divisor in |nK_{\overline{M}_{23}}| must contain Δ_{i} with multiplicity 16n when i = 1, 19n when i = 2, and (21 − i)n for i = 3, …, 9 or 11. The results for Δ_{1} and Δ_{2} are optimal since these multiplicities coincide with those in (14). Note that [Δ_{1}] = 2δ_{1}.

Next, one can show that certain geometric loci in M_{23} which are contained in all three Brill-Noether divisors, are contained in Σ as well. The method is based on the trivial observation that for a family f : X → B of stable curves of genus 23 with smooth general member, if B.K_{\overline{M}_{23}} < 0 (or equivalently, slope(X/B) = δ_{B}/λ_{B} > 13/2), then ϕ(B) ⊆ Σ, where ϕ : B → \overline{M}_{23}, ϕ(b) = [X_{b}], is the associated moduli map. We have that:

• One can fill up the d-gonal locus \overline{M}_{d}^{1} with families f : X → B of stable curves of genus g such that slope(X/B) is 8 + 4/g in the hyperelliptic case, and > 6 + 12/(g + 1) in the trigonal and tetragonal case (cf. [Sta]). For g = 23 it follows that M_{4}^{1} ⊆ Σ. Note that this result is not optimal if we believe the slope conjecture since we know that M_{8}^{1} ⊆ M_{12}^{1} ∩ M_{17}^{2} ∩ M_{20}^{3}. (The inclusion M_{8}^{1} ⊆ M_{20}^{3} is a particular case of a result from [CM].)

• We take a pencil of nodal plane curves of degree d with f assigned nodes in general position such that \binom{d−1}{2} − f = 23, and with b base points, where 4f + b = d^{2}. After blowing-up the base points, we have a pencil Y → ℙ^{1} with fibre [Y_{t}] ∈ \overline{M}_{d}^{2}. For this pencil λ_{ℙ^{1}} = χ(O_{Y}) + 23 − 1 = 23 and δ_{ℙ^{1}} = c_{2}(Y) + 88 = 91 + b + f. The condition δ_{ℙ^{1}}/λ_{ℙ^{1}} > 13/2 is satisfied precisely when d ≤ 10, hence taking into account that such pencils fill up M_{d}^{2}, we obtain that M_{10}^{2} ⊆ Σ. Note that M_{10}^{2} ⊆ M_{8}^{1}, and as mentioned above, the 8-gonal locus is contained in the intersection of the Brill-Noether divisors.

• In a similar fashion we can prove that M_{23,γ}(2), the locus of curves of genus 23 which are double coverings of curves of genus γ is contained in Σ for γ ≤ 5.

The fact that the slopes of other divisors on \overline{M}_{23} (or on \overline{M}_{g} for arbitrary g) consisting of curves with special geometric characterization, are larger than 6+12/(g + 1), lends further support to the slope hypothesis. In another paper we will compute the class of other divisors on \overline{M}_{23}: the closure in \overline{M}_{23} of the locus

{[C] ∈ M_{23} : C possesses a 𝔤_{13}^{1} with two different triple points},

and the closure of the locus

{[C] ∈ M_{23} : C has a 𝔤_{18}^{2} with a 5-fold point, i.e. ∃D ∈ C^{(5)} such that 𝔤_{18}^{2}(−D) = 𝔤_{13}^{1}}.

In each case we will show that the slope estimate holds.

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Korteweg-de Vries Institute for Mathematics, University of Amsterdam Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands

e-mail: farkas@wins.uva.nl