Brill-Noether Loci and the Gonality Stratification of M_{g}

GAVRIL FARKAS

1 Introduction

For an irreducible smooth projective complex curve C of genus g, the gonality defined as gon(C) = min{d ∈ ℤ_{≥1} : there exists a 𝔤_{d}^{1} on C} is perhaps the second most natural invariant: it gives an indication of how far C is from being rational, in a way different from what the genus does. For g ≥ 3 we consider the stratification of the moduli space M_{g} of smooth curves of genus g given by gonality:

M_{g,2}^{1} ⊆ M_{g,3}^{1} ⊆ … ⊆ M_{g,k}^{1} ⊆ … ⊆ M_{g},

where M_{g,k}^{1} := {[C] ∈ M_{g} : C has a 𝔤_{k}^{1}}. It is well-known that the k-gonal locus M_{g,k}^{1} is an irreducible variety of dimension 2g + 2k − 5 when k ≤ (g + 2)/2; when k ≥ [(g + 3)/2] one has that M_{g,k}^{1} = M_{g} (see for instance [AC]). The number [(g + 3)/2] is thus the generic gonality for curves of genus g.

For positive integers g, r and d, we introduce the Brill-Noether locus

M_{g,d}^{r} = {[C] ∈ M_{g} : C carries a 𝔤_{d}^{r}}.

The Brill-Noether Theorem (cf. [ACGH]) asserts that when the Brill-Noether number ρ(g, r, d) = g −(r+1)(g −d+r) is negative, the general curve of genus g has no 𝔤_{d}^{r}’s, hence in this case the locus M_{g,d}^{r} is a proper subvariety of M_{g}. We study the relative position of the loci M_{g,d}^{r} when r ≥ 3 and ρ(g, r, d) < 0 with respect to the gonality stratification of M_{g}. Typically, we would like to know the gonality of a ‘general’ point [C] ∈ M_{g,d}^{r}, or equivalently the gonality of a ‘general’ smooth curve C ⊆ ℙ^{r} of genus g and degree d. Since the geometry of the loci M_{g,d}^{r} is very messy (existence of many components, some nonreduced and/or not of expected dimension), we will content ourselves with computing gon(C) when [C] is a general point of a ‘nice’ component of M_{g,d}^{r} (i.e. a component which is generically smooth, of the expected dimension and with general point corresponding to a curve with a very ample 𝔤_{d}^{r}).

Our main result is the following:

Theorem 1 Let g ≥ 15 and d ≥ 14 be integers with g odd and d even, such that d^{2} > 8g, 4d < 3g + 12, d^{2} − 8g + 8 is not a square and either d ≤ 18 or g < 4d − 31. If

(d′, g′) ∈ {(d, g), (d + 1, g + 1), (d + 1, g + 2), (d + 2, g + 3)},

then there exists a regular component of the Hilbert scheme Hilb_{d′,g′,3} whose general point [C′] is a smooth curve such that gon(C′) =min(d′ − 4, [(g′ + 3)/2]).

Here by Hilb_{d,g,r} we denote the Hilbert scheme of curves C ⊆ ℙ^{r} with p_{a}(C) = g and deg(C) = d. A component of Hilb_{d,g,r} is said to be regular if its general point corresponds to a smooth irreducible curve C ⊆ ℙ^{r} such that the normal bundle N_{C/ℙ^{r}} satisfies H^{1}(C, N_{C/ℙ^{r}}) = 0. By standard deformation theory (cf. [Mod] or [Se]), a regular component of Hilb_{d,g,r} is generically smooth of the expected dimension χ(C, N_{C/ℙ^{r}}) = (r + 1)d − (r − 3)(g − 1). Note that for r = 3 the expected dimension of the Hilbert scheme is just 4d. We refer to Section 4 for a natural extension of Theorem 1 for curves in higher dimensional projective spaces.

As for the numerical conditions entering Theorem 1, we note that the inequality d^{2} > 8g ensures the existence of smooth curves C ⊆ ℙ^{3} with g(C) = g and deg(C) = d (see Section 2), 4d < 3g + 12 ⇔ ρ(g, 3, d) < 0 is just the condition that M_{g,d}^{3} is a proper subvariety of M_{g}, while the remaining requirements are mild technical conditions.

A remarkable application of Theorem 1 is a new proof of our result (cf. [Fa]):

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

We recall that for g ≥ 24 Harris, Mumford and Eisenbud proved (cf. [HM],[EH]) that M_{g} is of general type whereas for g ≤ 16, g ≠ 14 we have that κ(M_{g}) = −∞. The famous Slope Conjecture of Harris and Morrison predicts that M_{g} is uniruled for all g ≤ 22 (see [Mod]). Therefore the moduli space M_{23} appears as an intriguing transition case between two extremes: uniruledness and being of general type.

To put our Theorem 1 into perspective, let us note that for r = 2 we have the following result of M. Coppens (cf. [Co]): let ν : C → Γ be the normalization of a general, irreducible plane curve of degree d with δ = g − \binom{d−1}{2} nodes. Assume that 0 < δ < (d^{2} − 7d + 18)/2. Then gon(C) = d − 2.

This theorem says that there are no 𝔤_{d−3}^{1}’s on C. On the other hand a 𝔤_{d−2}^{1} is given by the lines through a node of Γ. The condition δ < (d^{2} − 7d + 18)/2 from the statement is equivalent with ρ(g, 1, d − 3) < 0. This is the range in which the problem is non-trivial: if ρ(g, 1, d − 3) ≥ 0, the Brill-Noether Theorem provides 𝔤_{d−3}^{1}’s on C.

For r ≥ 3 we might hope for a similar result. Let C ⊆ ℙ^{r} be a suitably general smooth curve of genus g and degree d, with ρ(g, r, d) < 0. We can always assume that d ≤ g − 1 (by duality 𝔤_{d}^{r} ↦ |K_{C} − 𝔤_{d}^{r}| we can always land in this range). One can expect that a 𝔤_{k}^{1} computing gon(C) is of the form 𝔤_{d}^{r}(−D) = {E − D : E ∈ 𝔤_{d}^{r}, E ≥ D} for some effective divisor D on C. Since the expected dimension of the variety of e-secant (r − 2)-plane divisors

V_{e}^{r−1}(𝔤_{d}^{r}) := {D ∈ C_{e} : dim 𝔤_{d}^{r}(−D) ≥ 1}

is 2r − 2 − e (cf. [ACGH]), we may ask whether C has finitely many (2r − 2)-secant (r − 2)-planes (and no (2r − 1)-secant (r − 2)-planes at all). This is known to be true for curves with general moduli, that is, when ρ(g, r, d) ≥ 0 (cf. [Hir]): for instance a smooth curve C ⊆ ℙ^{3} with general moduli has only finitely many 4-secant lines and no 5-secant lines. No such principle appears to be known for curves with special moduli. Definition: We call the number min(d − 2r + 2, [(g + 3)/2]) the expected gonality of a smooth nondegenerate curve C ⊆ ℙ^{r} of degree d and genus g.

One can approach such problems from a different angle: find recipes to compute the gonality of various classes of curves C ⊆ ℙ^{r}. Our knowledge in this respect is very scant: we know how to compute the gonality of extremal curves C ⊆ ℙ^{r} (that is, curves attaining the Castelnuovo bound, see [ACGH]) and the gonality of complete intersections in ℙ^{3} (cf. [Ba]): If C ⊆ ℙ^{3} is a smooth complete intersection of type (a, b) then gon(C) = ab − l, where l is the degree of a maximal linear divisor on C. Hence an effective divisor D ⊆ C computing gon(C) is residual to a linear divisor of degree l in a plane section of C. Acknowledgments: This paper is part of my thesis written at the Universiteit van Amsterdam. The help of my advisor Gerard van der Geer, and of Joe Harris, is gratefully acknowledged.

2 Linear systems on K3 surfaces in ℙ^{r}

We will construct smooth curves C ⊆ ℙ^{r} having the expected gonality starting with sections of smooth K3 surfaces. We recall a few basic facts about linear systems on K3 surfaces (cf. [SD]).

Let S be a smooth K3 surface. For an effective divisor D ⊆ S, we have h^{1}(S, D) = h^{0}(D, O_{D})−1. If C ⊆ S is an irreducible curve then H^{1}(S, C) = 0, and by Riemann-Roch we have that dim|C| = 1 + C^{2}/2 = p_{a}(C). In particular C^{2} ≥ −2 for every irreducible curve C. Moreover we have equivalences

C^{2} = −2 ⟺ dim|C| = 0 ⟺ C is a smooth rational curve and

C^{2} = 0 ⟺ dim|C| = 1 ⟺ p_{a}(C) = 1.

For a K3 surface one also has a ‘strong Bertini’ Theorem (cf. [SD]):

Proposition 2.1 Let L be a line bundle on a K3 surface S such that |L| ≠ ∅. Then |L| has no base points outside its fixed components. Moreover, if bs|L| = ∅ then either

• L^{2} > 0, h^{1}(S, L) = 0 and the general member of |L| is a smooth, irreducible curve of genus L^{2}/2 + 1, or

• L^{2} = 0 and L = O_{S}(kE), where k ∈ ℤ_{≥1}, E ⊆ S is an irreducible curve with p_{a}(E) = 1. We have that h^{0}(S, L) = k + 1, h^{1}(S, L) = k − 1 and all divisors in |L| are of the form E_{1} + ⋯ + E_{k} with E_{i} ∼ E.

We are interested in space curves sitting on K3 surfaces and the starting point is Mori’s Theorem (cf. [Mo]): if d > 0, g ≥ 0, there is a smooth curve C ⊆ ℙ^{3} of degree d and genus g, lying on a smooth quartic surface S, if and only if (1) g = d^{2}/8+1, or (2) g < d^{2}/8 and (d, g) ≠ (5, 3). Moreover, we can choose S such that Pic(S) = ℤH = ℤ(4/d)C in case (1) and such that Pic(S) = ℤH ⊕ ℤC, with H^{2} = 4, C^{2} = 2g − 2 and H · C = d, in case (2). In each case H denotes a plane section of S. Note that from the Hodge Index Theorem one has the necessary condition (C · H)^{2} − H^{2}C^{2} = d^{2} − 8(g − 1) ≥ 0.

Mori’s result has been extended by Rathmann to curves in higher dimensional projective spaces (cf. [Ra], see also [Kn]): For integers d > 0, g > 0 and r ≥ 3 such that d^{2} ≥ 4g(r − 1) + (r − 1)^{2}, there exists a smooth K3 surface S ⊆ ℙ^{r} of degree 2r − 2 and a smooth curve C ⊆ S of genus g and degree d such that Pic(S) = ℤH ⊕ ℤC, where H is a hyperplane section of S.

We will repeatedly use the following simple observation:

Proposition 2.2 Let S ⊆ ℙ^{r} be a smooth K3 surface of degree 2r − 2 with a smooth curve C ⊆ S such that Pic(S) = ℤH ⊕ ℤC and assume that S has no (−2) curves. A divisor class D on S is effective if and only if D^{2} ≥ 0 and D · H > 2.

Remark: If S ⊆ ℙ^{r} is a smooth K3 surface of degree 2r − 2 with Picard number 2 as above, S has no (−2) curves when the equation

(r − 1)m^{2} + mnd + (g − 1)n^{2} = −1  (1)

has no solutions m, n ∈ ℤ. This is the case for instance when d is even and g and r are odd. Furthermore, a necessary condition for S to have genus 1 curves is that d^{2} −4(g −1)(r−1) is a square.

3 Brill-Noether special linear series on curves on K3 surfaces

The first important result in the study of special linear series on curves lying on K3 surfaces was Lazarsfeld’s proof of the Brill-Noether-Petri Theorem (cf. [Laz]). He noticed that there is no Brill-Noether type obstruction to embed a curve in a K3 surface: if C_{0} ⊆ S is a smooth curve of genus g ≥ 2 on a K3 surface such that Pic(S) = ℤC_{0}, then the general curve C ∈ |C_{0}| satisfies the Brill-Noether-Petri Theorem, that is, for any line bundle A on C, the Petri map μ_{0}(C, A) : H^{0}(C, A) ⊗ H^{0}(C, K_{C} ⊗ A^{∨}) → H^{0}(C, K_{C}) is injective. We mention that Petri’s Theorem implies (trivially) the Brill-Noether Theorem.

The general philosophy when studying linear series on a K3-section C ⊆ S of genus g ≥ 2, is that the type of a Brill-Noether special 𝔤_{d}^{r} often does not depend on C but only on its linear equivalence class in S, i.e. a 𝔤_{d}^{r} on C with ρ(g, r, d) < 0 is expected to propagate to all smooth curves C′ ∈ |C|. This expectation, in such generality, is perhaps a bit too optimistic, but it was proved to be true for the Clifford index of a curve (see [GL]): for C ⊆ S a smooth K3-section of genus g ≥ 2, one has that Cliff(C′) = Cliff(C) for every smooth curve C′ ∈ |C|. Furthermore, if Cliff(C) < [(g − 1)/2] (the generic value of the Clifford index), then there exists a line bundle L on S such that for all smooth C′ ∈ |C| the restriction L_{|C′} computes Cliff(C′). Recall that the Clifford index of a curve C of genus g is defined as

Cliff(C) := min{Cliff(D) : D ∈ Div(C), h^{0}(D) ≥ 2, h^{1}(D) ≥ 2},

where for an effective divisor D on C, we have Cliff(D) = deg(D)−2(h^{0}(D)−1). Note that in the definition of Cliff(C) the condition h^{1}(D) ≥ 2 can be replaced with deg(D) ≤ g −1. Another invariant of a curve is the Clifford dimension of C defined as

Cliff-dim(C) := min{r ≥ 1 : ∃𝔤_{d}^{r} on C with d ≤ g − 1, such that d − 2r = Cliff(C)}.

Curves with Clifford dimension ≥ 2 are rare: smooth plane curves are precisely the curves of Clifford dimension 2, while curves of Clifford dimension 3 occur only in genus 10 as complete intersections of two cubic surfaces in ℙ^{3}.

Harris and Mumford during their work in [HM] conjectured that the gonality of a K3-section should stay constant in a linear system: if C ⊆ S carries an exceptional 𝔤_{d}^{1} then every smooth C′ ∈ |C| carries an equally exceptional 𝔤_{d}^{1}. This conjecture was later disproved by Donagi and Morrison (cf. [DMo]). They came up with the following counterexample: let π : S → ℙ^{2} be a K3 surface, double cover of ℙ^{2} branched along a smooth sextic and let L = π^{∗}O_{ℙ^{2}}(3). The genus of a smooth C ∈ |L| is 10. The general C ∈ |L| carries a very ample 𝔤_{6}^{2}, hence gon(C) = 5. On the other hand, any curve in the codimension 1 linear system |π^{∗}H^{0}(ℙ^{2}, O_{ℙ^{2}}(3))| is bielliptic, therefore has gonality 4. Under reasonable assumptions this turns out to be the only counterexample to the Harris-Mumford conjecture. Ciliberto and Pareschi proved that if C ⊆ S is such that |C| is base-point-free and ample, then either gon(C′) = gon(C) for all smooth C′ ∈ |C|, or (S, C) are as in the previous counterexample (cf. [CilP]).

Although gon(C) can drop as C varies in a linear system, base-point-free 𝔤_{d}^{1}’s on K3-sections do propagate:

Proposition 3.1 (Donagi-Morrison) Let S be a K3 surface, C ⊆ S a smooth, nonhyperelliptic curve and |Z| a complete, base-point-free 𝔤_{d}^{1} on C such that ρ(g, 1, d) < 0. Then there is an effective divisor D ⊆ S such that:

• h^{0}(S, D) ≥ 2, h^{0}(S, C − D) ≥ 2, deg_{C}(D_{|C}) ≤ g − 1.

• Cliff(C′, D_{|C′}) ≤ Cliff(C, Z), for any smooth C′ ∈ |C|.

• There is Z_{0} ∈ |Z|, consisting of distinct points such that Z_{0} ⊆ D ∩ C.

Throughout this paper, for a smooth curve C we denote, as usual, by W_{d}^{r}(C) the scheme whose points are line bundles A ∈ Pic^{d}(C) with h^{0}(C, A) ≥ r + 1, and by G_{d}^{r}(C) the scheme parametrizing 𝔤_{d}^{r}’s on C.

4 The gonality of curves in ℙ^{r}

For a wide range of d, g and r we construct curves C ⊆ ℙ^{r} of degree d and genus g having the expected gonality. We start with a case when we can realize our curves as sections of K3 surfaces.

Theorem 3 Let r ≥ 3, d ≥ r^{2} + r and g ≥ 0 be integers such that ρ(g, r, d) < 0 and with d^{2} > 4(r − 1)(g + r − 2) when r ≥ 4 while d^{2} > 8g when r = 3. Let us assume moreover that 0 and −1 are not represented by the quadratic form

Q(m, n) = (r − 1)m^{2} + mnd + (g − 1)n^{2}, m, n ∈ ℤ.

Then there exists a smooth curve C ⊆ ℙ^{r} of degree d and genus g such that gon(C) = min(d−2r +2, [(g +3)/2]). If gon(C) = d−2r +2 < [(g +3)/2] then dim W_{d−2r+2}^{1}(C) = 0 and every 𝔤_{d−2r+2}^{1} is given by the hyperplanes through a (2r − 2)-secant (r − 2)-plane.

Proof: By Rathmann’s Theorem there exists a smooth K3 surface S ⊆ ℙ^{r} with deg(S) = 2r − 2 and C ⊆ S a smooth curve of degree d and genus g such that Pic(S) = ℤH ⊕ ℤC, where H is a hyperplane section. The conditions d, g and r are subject to, ensure that S does not contain (−2) curves or genus 1 curves.

We prove first that Cliff-dim(C) = 1. It suffices to show that C ⊆ S is an ample divisor, because then by using Prop.3.3 from [CilP] we obtain that either Cliff-dim(C) = 1 or C is a smooth plane sextic, g = 10 and (S, C) are as in Donagi-Morrison’s example (then Cliff-dim(C) = 2). The latter case obviously does not happen.

We prove that C · D > 0 for any effective divisor D ⊆ S. Let D ∼ mH + nC, with m, n ∈ ℤ, such a divisor. Then D^{2} = (2r − 2)m^{2} + 2mnd + n^{2}(2g − 2) ≥ 0 and D · H = (2r − 2)m + dn > 2. The case m ≤ 0, n ≤ 0 is impossible, while the case m ≥ 0, n ≥ 0 is trivial. Let us assume m > 0, n < 0. Then D · C = md + n(2g − 2) > −n(d^{2}/(2r − 2) − 2g + 2) + d/(r − 1) > 0, because d^{2}/(2r − 2) > 2g. In the remaining case m < 0, n > 0 we have that nD · C ≥ −mD · H > 0, so C is ample by Nakai-Moishezon.

Our assumptions imply that d ≤ g −1, so O_{C}(1) is among the line bundles from which Cliff(C) is computed. We get thus the following estimate on the gonality of C:

gon(C) = Cliff(C) + 2 ≤ Cliff(C, H_{|C}) + 2 = d − 2r + 2,

which yields gon(C) ≤ min(d − 2r + 2, [(g + 3)/2]).

For the rest of the proof let us assume that gon(C) < [(g + 3)/2]. We will then show that gon(C) = d−2r +2. Let |Z| be a complete, base point free pencil computing gon(C). By applying Prop.3.1, there exists an effective divisor D ⊆ S satisfying

h^{0}(S, D) ≥ 2, h^{0}(S, C−D) ≥ 2, deg(D_{|C}) ≤ g−1, gon(C) = Cliff(D_{|C})+2 and Z ⊆ D∩C.

We consider the exact cohomology sequence:

0 → H^{0}(S, D − C) → H^{0}(S, D) → H^{0}(C, D_{|C}) → H^{1}(S, D − C).

Since C−D is effective and ≁ 0, one sees that D−C cannot be effective, so H^{0}(S, D−C) = 0. The surface S does not contain (−2) curves, so |C − D| has no fixed components; the equation (C −D)^{2} = 0 has no solutions, therefore (C −D)^{2} > 0 and the general element of |C−D| is smooth and irreducible. Then it follows that H^{1}(S, D−C) = H^{1}(S, C−D)^{∨} = 0. Thus H^{0}(S, D) = H^{0}(C, D_{|C}) and

gon(C) = 2 + Cliff(D_{|C}) = 2 + D · C − 2 dim|D| = D · C − D^{2}.

We consider the following family of effective divisors

A := {D ∈ Div(S) : h^{0}(S, D) ≥ 2, h^{0}(S, C − D) ≥ 2, C · D ≤ g − 1}.

Since we already know that d−2r+2 ≥ gon(C) ≥ α, where α = min{D·C −D^{2} : D ∈ A}, we are done if we prove that α ≥ d − 2r + 2. Take D ∈ A such that D ∼ mH + nC, m, n ∈ ℤ. The conditions D^{2} > 0, D · C ≤ g − 1 and 2 < D · H < d − 2 (use Prop.2.2 for the last inequality) can be rewritten as

(r−1)m^{2}+mnd+n^{2}(g−1) > 0 (i), 2 < (2r−2)m+nd < d−2 (ii), md+(2n−1)(g−1) ≤ 0 (iii).

We have to prove that for any D ∈ A the following inequality holds

f(m, n) = D·C −D^{2} = −(2r−2)m^{2} +m(d−2nd)+(n−n^{2})(2g−2) ≥ f(1, 0) = d−2r+2.

We solve this standard calculus problem. Denote by

a := \frac{d + \sqrt{d^{2} − 4(r − 1)(g − 1)}}{2r − 2} and b := \frac{d − \sqrt{d^{2} − 4(r − 1)(g − 1)}}{2r − 2} .

We dispose first of the case n < 0. Assuming n < 0, from (i) we have that either m < −bn or m > −an. If m < −bn from (ii) we obtain that 2 < n(d − (2r − 2)b) < 0, because n < 0 and d − (2r − 2)b = \sqrt{d^{2} − 4(r − 1)(g − 1)} > 0, so we have reached a contradiction.

We assume now that n < 0 and m > −an. From (iii) we get that m ≤ (g−1)(1−2n)/d. If −an > (g − 1)(1 − 2n)/d we are done because there are no m, n ∈ ℤ satisfying (i), (ii) and (iii), while in the other case for any D ∈ A with D ∼ mH + nC, one has the inequality

f(m, n) > f(−an, n) = \frac{(d^{2} − 4(r − 1)(g − 1)) + d\sqrt{d^{2} − 4(r − 1)(g − 1)}}{2r − 2}(−n).

When r ≥ 4 since we assume that \sqrt{d^{2} − 4(r − 1)(g − 1)} ≥ 2r − 2, it immediately follows that f(m, n) ≥ d > d − 2r + 2. In the case r = 3 when we only have the weaker assumption d^{2} > 8g, we still get that f(−an, n) > d − 4 unless n = −1 and d^{2} − 8g < 8. In this last situation we obtain m ≥ (d + 4)/4 so f(m, −1) ≥ f((d + 4)/4, −1) > d − 4.

The case n > 0 can be treated rather similarly. From (i) we get that either m < −an or m > −bn. The first case can be dismissed immediately. When m > −bn we use that for any D ∈ A with D ∼ mH + nC,

f(m, n) ≥ min{f(−(g − 1)(2n − 1)/d, n), max{f(−bn, n), f((2 − nd)/(2r − 2), n)}}.

Elementary manipulations give that

f(−(g − 1)(2n − 1)/d, n) = (g − 1)/2 [(2n − 1)^{2}(d^{2} − 4(r − 1)(g − 1))/d^{2} + 1] ≥ d − 2r + 2

(use only that d ≤ g − 1 and d^{2} > 4(r − 1)g, so we cover both cases r = 3 and r ≥ 4 at once). Note that in the case n > 0 we have equality if and only if n = 1, m = −1 and d = g − 1.

Moreover f(−bn, n) = n(2g − 2 − bd) ≥ 2g − 2 − bd and 2g − 2 − bd > d − 2r + 2 ⇔ 2r − 2 < \sqrt{d^{2} − 4(r − 1)(g − 1)} < d − 2r + 2. When this does not happen we proceed as follows: if \sqrt{d^{2} − 4(r − 1)(g − 1)} ≥ d − 2r + 2 then if n = 1 we have that m > −b ≥ −1, that is m ≥ 0, but this contradicts (ii). When n ≥ 2, we have f((2 − nd)/(2r − 2), n) = [(d^{2} − 4(r − 1)(g − 1))(n^{2} − n) + (2d − 4)]/(2r − 2) > d − 2r + 2. Finally, the remaining possibility 2r − 2 ≥ \sqrt{d^{2} − 4(r − 1)(g − 1)} does not occur when r ≥ 4 while in the case r = 3 we either have f(−bn, n) > d − 4 or else n = 1 and then m > (−d + 4)/4 hence f(m, 1) > f((−d + 4)/4, 1) = d − 4.

All this leaves us with the case n = 0, when f(m, 0) = −(2r − 2)m^{2} + md. Clearly f(m, 0) ≥ f(1, 0) for all m complying with (i),(ii) and (iii).

Thus we proved that gon(C) = d − 2r + 2. We have equality D · C − D^{2} = d − 2r + 2 where D ∈ A, if and only if D = H or in the case d = g − 1 also when D = C − H. The latter possibility can be ruled out since d = g − 1 is not compatible with the assumptions d ≥ r^{2} + r and d − 2r + 2 < [(g + 3)/2]. Therefore we can always assume that the divisor on S cutting a 𝔤_{d−2r+2}^{1} on C is the hyperplane section of S. Since Z ⊆ H ∩C, if we denote by Δ the residual divisor of Z in H ∩ C, we have that h^{0}(C, H_{|C} − Δ) = 2, so Δ spans a ℙ^{r−2} hence |Z| is given by the hyperplanes through the (2r − 2)-secant (r − 2)-plane ⟨Δ⟩. This shows that every pencil computing gon(C) is given by the hyperplanes through a (2r − 2)-secant (r − 2)-plane.

There are a few ways to see that C has only finitely many (2r−2)-secant (r−2)-planes. The shortest is to invoke Theorem 3.1 from [CilP]: since gon(C′) = d − 2r + 2 is constant as C′ varies in |C|, for the general smooth curve C′ ∈ |C| one has dim W_{d−2r+2}^{1}(C′) = 0. □

Remarks: 1. Keeping the assumptions and the notations of Theorem 3 we note that when d − 2r + 2 < [(g + 3)/2] the linear system |C| is (d − 2r − 1)-very ample, i.e. for any 0-dimensional subscheme Z ⊆ S of length ≤ d − 2r the map H^{0}(S, C) → H^{0}(S, C ⊗ O_{Z}) is surjective. Indeed, by applying Theorem 2.1 from [BS] if |C| is not (d − 2r − 1)-very ample, there exists an effective divisor D on S such that C − 2D is ℚ-effective and

C · D − (d − 2r) ≤ D^{2} ≤ C · D/2 < d − 2r,

hence C ·D−D^{2} ≤ d−2r. On the other hand clearly D ∈ A, thus C ·D−D^{2} ≥ d−2r+2, a contradiction.

2. One can find quartic surfaces S ⊆ ℙ^{3} containing a smooth curve C of degree d and genus g in the case g = d^{2}/8 + 1 (which is outside the range Theorem 3 deals with). Then d = 4m, g = 2m^{2} + 1 with m ≥ 1 and C is a complete intersection of type (4, m). For such a curve, gon(C) = d − l, where l is the degree of a maximal linear divisor on C (cf. [Ba]). If S is sufficiently general so that it contains no lines, by Bezout, C cannot have 5-secant lines so gon(C) = d − 4 in this case too.

When r = 3 we want to find out when the curves constructed in Theorem 3 correspond to smooth points of Hilb_{d,g,3}. We have the following:

Proposition 4.1 Let C ⊆ S ⊆ ℙ^{3} be a smooth curve sitting on a quartic surface such that Pic(S) = ℤH ⊕ ℤC with H being a plane section and assume furthermore that S contains no (−2) curves. Then H^{1}(C, N_{C/ℙ^{3}}) = 0 if and only if d ≤ 18 or g < 4d − 31.

Proof: We use the exact sequence

0 ⟶ N_{C/S} ⟶ N_{C/ℙ^{3}} ⟶ N_{S/ℙ^{3}} ⊗ O_{C} ⟶ 0,  (2)

where N_{S/ℙ^{3}} ⊗ O_{C} = O_{C}(4) and N_{C/S} = K_{C}. We claim that there is an isomorphism H^{1}(C, N_{C/ℙ^{3}}) = H^{1}(C, O_{C}(4)). Suppose this is not the case. Then the injective map H^{1}(C, K_{C}) → H^{1}(C, N_{C/ℙ^{3}}) provides a section σ ∈ H^{0}(N_{C/ℙ^{3}}^{∨} ⊗ K_{C}) which yields a splitting of the dual of the exact sequence (2), hence (2) is split as well. Using a result from [GH, p.252] we obtain that C is a complete intersection with S. This is clearly a contradiction. Therefore one has H^{1}(C, N_{C/ℙ^{3}}) = H^{1}(C, O_{C}(4)).

We have isomorphisms H^{1}(C, 4H_{|C}) = H^{2}(S, 4H − C) = H^{0}(S, C − 4H)^{∨}. According to Prop.2.2 the divisor C−4H is effective if and only if (C−4H)^{2} ≥ 0 and (C−4H)·H > 2, from which the conclusion follows. □

We need to determine the gonality of nodal curves not of compact type and which consist of two components meeting at a number of points. We have the following result:

Proposition 4.2 Let C = C_{1} ∪_{Δ} C_{2} be a quasi-transversal union of two smooth curves C_{1} and C_{2} meeting at a finite set Δ. Denote by g_{1} = g(C_{1}), g_{2} = g(C_{2}), δ = card(Δ). Let us assume that C_{1} has only finitely many pencils 𝔤_{d}^{1}, where δ ≤ d and that the points of Δ do not occur in the same fibre of one of these pencils. Then gon(C) ≥ d + 1. Moreover if gon(C) = d + 1 then either (1) C_{2} is rational and there is a degree d map f_{1} : C_{1} → ℙ^{1} and a degree 1 map f_{2} : C_{2} → ℙ^{1} such that f_{1 |Δ} = f_{2 |Δ}, or (2) there is a 𝔤_{d+1}^{1} on C_{1} containing Δ in a fibre.

Proof: Let us assume that C is k-gonal, that is, a limit of smooth k-gonal curves. If g = g_{1} + g_{2} + δ − 1, we consider the space \overline{H}_{g,k} of Harris-Mumford admissible coverings of degree k and we denote by π : \overline{H}_{g,k} → \overline{M}_{g} the proper map sending a covering to the stable model of its domain (see [HM]). Since [C] ∈ \overline{M}_{g,k}^{1} = Im(π), it follows that there exists a semistable curve C′ whose stable model is C and a degree k admissible covering f : C′ → Y, where Y is a semistable curve of arithmetic genus 0. We thus have that f^{−1}(Y_{sing}) = C′_{sing} and if p ∈ C′_{1} ∩ C′_{2} with C′_{1} and C′_{2} components of C′, then f(C′_{1}) and f(C′_{2}) are distinct components of Y and the ramification indices at the point p of the restrictions f_{|C′_{1}} and f_{|C′_{2}} are the same.

We have that C′ = C_{1} ∪ C_{2} ∪ R_{1} ∪ … ∪ R_{δ}, where for 1 ≤ i ≤ δ the curve R_{i} is a (possibly empty) destabilizing chain of ℙ^{1}’s inserted at the nodes of C. Let us denote {p_{i}} = C_{1} ∩ R_{i} and {q_{i}} = C_{2} ∩ R_{i}; if R_{i} = ∅ then we take p_{i} = q_{i} ∈ Δ ⊆ C.

We first show that k ≥ d + 1. Suppose k ≤ d. Since C_{1} has no 𝔤_{d−1}^{1}’s it follows that k = d and that f^{−1}f(C_{1}) = C_{1}. If there were distinct points p_{i} and p_{j} such that f(p_{i}) ≠ f(p_{j}), then f(R_{i}) ≠ f(R_{j}) and the image curve Y would no longer have genus 0. Therefore f(p_{i}) = f(p_{j}) for all i, j ∈ {1, …, δ}, that is Δ appears in the fibre of a 𝔤_{d}^{1} on C_{1}, a contradiction.

Assume now that k = d + 1. Then either deg(f_{|C_{1}}) = d or deg(f_{|C_{1}}) = d + 1. If deg(f_{|C_{1}}) = d + 1, then again f^{−1}f(C_{1}) = C_{1} and by the same reasoning f maps all the p_{i}’s to the same point and this yields case (2) from the statement of the Proposition. If deg(f_{|C_{1}}) = d then f^{−1}f(C_{1}) = C_{1} ∪ D, where D is a smooth rational curve mapped isomorphically to its image via f. If D = C_{2} then the condition that the dual graph of Y is a tree implies that f(p_{i}) = f(q_{i}) for all i and this yields case (1) from the statement. Finally, if D ≠ C_{2} then f(C_{1}) ≠ f(C_{2}). We know that there are 1 ≤ i < j ≤ δ such that f(p_{i}) ≠ f(p_{j}). The image f(C_{2}) belongs to a chain R of ℙ^{1}’s such that either R ∩ f(C_{1}) = {f(p_{i})} or R ∩ f(C_{1}) = {f(p_{j})}. In the former case f(p) = f(p_{i}) for all p ∈ Δ − {p_{j}} while in the latter case f(p) = f(p_{j}) for all p ∈ Δ − {p_{i}}. In each case by adding a base point we obtain a 𝔤_{d+1}^{1} on C_{1} containing Δ in a fibre. □

Theorem 3 provides curves C ⊆ ℙ^{3} of expected gonality when d is even and g is odd (equation (1) has no solutions in this case). Naturally, we would like to have such curves when d and g have other parities as well. We will achieve this by attaching to a ‘good’ curve of expected gonality either a 2 or 3-secant line or a 4-secant conic.

Theorem 1 Let g ≥ 15 and d ≥ 14 be integers with g odd and d even, such that d^{2} > 8g, 4d < 3g + 12, d^{2} − 8g + 8 is not a square and either d ≤ 18 or g < 4d − 31. If

(d′, g′) ∈ {(d, g), (d + 1, g + 1), (d + 1, g + 2), (d + 2, g + 3)},

then there exists a regular component of Hilb_{d′,g′,3} with general point [C′] a smooth curve such that gon(C′) = min(d′ − 4, [(g′ + 3)/2]).

Proof: For d and g as in the statement we know by Theorem 3 and by Prop.4.1 that there exists a smooth nondegenerate curve C ⊆ ℙ^{3} of degree d and genus g, with gon(C) = min(d−4, [(g +3)/2]) and H^{1}(C, N_{C/ℙ^{3}}) = 0. We can also assume that C sits on a smooth quartic surface S and Pic(S) = ℤH ⊕ ℤC. Moreover, in the case d − 4 < [(g + 3)/2] the curve C has only finitely many 𝔤_{d−4}^{1}’s, all given by planes through a 4-secant line.

i) Let us settle first the case (d′, g′) = (d + 1, g + 1). Take p, q ∈ C general points, L = \overline{pq} ⊆ ℙ^{3} and X := C ∪ L. By applying Lemma 1.2 from [BE], we know that H^{1}(X, N_{X}) = 0 and the curve X is smoothable in ℙ^{3}, that is, there exists a flat family of curves {X_{t}} in ℙ^{3} over a smooth and irreducible base, with the general fibre X_{t} smooth while the special fibre X_{0} is X. If d − 4 < [(g + 3)/2], then since C has only finitely many 𝔤_{d−4}^{1}’s, by applying Prop.4.2 we get that gon(X) = d − 3. In the case d − 4 ≥ [(g + 3)/2] we just notice that gon(X) ≥ gon(C) = [(g′ + 3)/2].

ii) Next we tackle the case (d′, g′) = (d + 1, g + 2). Assume first that d − 4 < [(g +3)/2] ⇔ d′ −4 < [(g′ +3)/2]. We apply Lemma 1.2 from [BE] to a curve X := C ∪L, where L is a suitable trisecant line to C. In order to conclude that X is smoothable in ℙ^{3} and that H^{1}(X, N_{X}) = 0, we have to make sure that the trisecant line L = \overline{pqq′} with p, q, q′ ∈ C can be chosen in such a way that

L, T_{p}(C), T_{q}(C) and T_{q′}(C) do not all lie in the same plane.  (3)

We claim that when C ∈ |C| is general in its linear system, at least one of its trisecants satisfies (3). Suppose not. Then for every smooth curve C ∈ |C| and for every trisecant line L to C condition (3) fails.

We consider a 0-dimensional subscheme Z ⊆ S where Z = p + q + q′ + u + u′, with p, q, q′ ∈ S being collinear points while u and u′ are general infinitely near points to q and q′ respectively. The linear system |C| is at least 5-very ample (cf. Remark 1), hence a general curve C ∈ |C − Z| is smooth and possesses a trisecant line for which (3) holds, a contradiction.

Since the scheme of trisecants to a space curve is of pure dimension 1, it follows that for a general curve C ∈ |C|, through a general point p ∈ C there passes a trisecant line L for which (3) holds. We have that X := C ∪ L is smoothable in ℙ^{3} and H^{1}(X, N_{X}) = 0. We conclude that gon(X) = d − 3 by proving that there is no 𝔤_{d−4}^{1} on C containing L ∩ C in a fibre.

If C ∈ |C| is general, any line in ℙ^{3} (hence also a 4-secant line to C) can meet only finitely many trisecants. Indeed, assuming that m ⊆ ℙ^{3} is a line meeting infinitely many trisecants, we consider the correspondence

T = \{(p, t) ∈ C × m : \overline{pt} is a trisecant to C\}

and the projections π_{1} : T → C and π_{2} : T → m. If π_{2} is surjective, then Nm_{π_{1}}(π_{2}) yields a 𝔤_{3}^{1} on C, a contradiction. If π_{2} is not surjective then there exists a point t ∈ ℙ^{3} such that \overline{pt} is a trisecant to C for each p ∈ C. This possibility cannot occur for a general C ∈ |C|: Otherwise we take general points t ∈ ℙ^{3} and p, p′ ∈ S and if we denote

B := \{C ∈ |C| : p, p′ ∈ C and \overline{tx} is a trisecant to C for each x ∈ C\},

we have that dim B ≥ g − 5. On the other hand since \overline{tp} and \overline{tp′} are trisecants for all curves C ∈ B, there must be a 0-dimensional subscheme Z ⊆ (\overline{tp} ∪ \overline{tp′}) ∩ S of length 6 such that B ⊆ |C − Z|, hence dim B ≤ dim|C − Z| = g − 6 (use again that |C| is 5-very ample), a contradiction. In this way the case d − 4 < [(g + 3)/2] is settled.

When d − 4 ≥ [(g + 3)/2] we apply Theorem 3 to obtain a smooth curve C_{1} ⊆ ℙ^{3} of degree d and genus g + 2 such that gon(C_{1}) = (g + 5)/2 and H^{1}(C_{1}, N_{C_{1}}) = 0. We take X_{1} := C_{1} ∪ L_{1} with L_{1} being a general 1-secant line to C_{1}. Then X_{1} is smoothable and gon(X_{1}) = gon(C_{1}) = (g + 5)/2.

iii) Finally, we turn to the case (d′, g′) = (d + 2, g + 3). Take H ⊆ ℙ^{3} a general plane meeting C in d distinct points in general linear position and pick 4 of them: p_{1}, p_{2}, p_{3}, p_{4} ∈ C ∩ H. Choose Q ⊆ H a general conic such that Q ∩ C = {p_{1}, p_{2}, p_{3}, p_{4}}. Theorem 5.2 from [Se] ensures that X := C ∪ Q is smoothable in ℙ^{3} and H^{1}(X, N_{X}) = 0.

Assume first that d′ −4 ≤ [(g′ +3)/2]. We claim that gon(X) ≥ gon(C)+2. According to Prop.4.2 the opposite could happen only in 2 cases: a) There exists a 𝔤_{d−3}^{1} on C, say |Z|, such that |Z|(−p_{1} − p_{2} − p_{3} − p_{4}) ≠ ∅. b) There exists a degree d − 4 map f : C → ℙ^{1} and a degree 1 map f′ : Q → ℙ^{1} such that f(p_{i}) = f′(p_{i}), for i = 1, …, 4.

Assume that a) does happen. We denote by U = {D ∈ C_{4} : |O_{C}(1)|(−D) ≠ ∅} the irreducible 3-fold of divisors of degree 4 spanning a plane and also consider the correspondence

Σ = {(L, D) ∈ W_{d−3}^{1}(C) × U : |L|(−D) ≠ ∅},

with the projections π_{1} : Σ → W_{d−3}^{1}(C) and π_{2} : Σ → U. We know that π_{2} is dominant, hence dim Σ ≥ 3 and therefore dim W_{d−3}^{1}(C) ≥ 2.

If ρ(g, 1, d − 3) < 0 by Prop.3.1 we get that every base-point-free 𝔤_{d−3}^{1} on C is cut out by a divisor D on S such that D ∈ A (see the proof of Theorem 3 for this notation) and C · D − D^{2} = Cliff(C, D_{|C}) + 2 ≤ d − 3, hence C · D − D^{2} ≤ d − 4 for parity reasons. As pointed out at the end of the proof of Theorem 3 this forces D ∼ H, that is, all base-point-free 𝔤_{d−3}^{1}’s on C are given by planes through a trisecant line. Thus C has ∞^{2} trisecants, a contradiction.

If ρ(g, 1, d − 3) ≥ 0, then g = 2d − 9 and we can assume that there is L ∈ π_{1}(Σ) such that |O_{C}(1) − L| ≠ ∅. The map π_{1} is either generically finite hence dim W_{d−4}^{1}(C) ≥ dim W_{d−3}^{1}(C) − 2 ≥ 1 (cf. [FHL]), a contradiction, or otherwise π_{1} has fibre dimension 1. This is possible only when there is a component A of W_{d−3}^{1}(C) with dim(A) ≥ 2 and such that the general L ∈ A satisfies |O_{C}(1) − L| ≠ ∅ and every L ∈ A has nonordinary ramification so that the monodromy of each 𝔤_{d−3}^{1} is not the full symmetric group. Applying again [FHL] there is L ∈ W_{d−4}^{1}(C) such that {L} + W_{1}^{0}(C) ⊆ A, in particular L has non-ordinary ramification too. It is easy to see that this contradicts the (d − 7)-very ampleness of |C| asserted by Remark 1.

We now rule out case b). Suppose that b) does happen and denote by L ⊆ ℙ^{3} the 4secant line corresponding to f. Let {p} = L∩H, and pick l ⊆ H a general line. As Q was a general conic through p_{1}, …, p_{4} we may assume that p ∉ Q. The map f′ : Q → l is (up to a projective isomorphism of l) the projection from a point q ∈ Q, while f(p_{i}) = \overline{p_{i}p} ∩ l, for i = 1, …, 4. By Steiner’s Theorem from classical projective geometry, the condition (f(p_{1})f(p_{2})f(p_{3})f(p_{4})) = (f′(p_{1})f′(p_{2})f′(p_{3})f′(p_{4})) is equivalent with p_{1}, p_{2}, p_{3}, p_{4}, p and q being on a conic, a contradiction since p ∉ Q.

Finally, when d′ − 4 > [(g′ + 3)/2], we have to show that gon(X) ≥ gon(C) + 1. We note that dim G_{(g+3)/2}^{1}(C) = 1 (for any curve one has the inequality dim G1_{gon} ≤ 1). By taking H ∈ (ℙ^{3})^{∨} general enough, we obtain that p_{1}, …, p_{4} do not occur in the same fibre of a 𝔤_{(g+3)/2}^{1}. □

Remark: Theorem 1 can be viewed as a non-containment relation M_{g′,d′}^{3} ⊈ M_{g′,d′−5}^{1} between different Brill-Noether loci when d′ and g′ are as in Theorem 1 and moreover d′ −4 ≤ [(g′ +3)/2]. We can turn this problem on its head and ask the following question: given g and k such that k < (g + 2)/2, when is it true that the general k-gonal curve of genus g has no other linear series 𝔤_{d}^{r} with ρ(g, r, d) < 0, that is, the pencil computing the gonality is the only Brill-Noether exceptional linear series?

In [Fa2] we prove using limit linear series the following result: fix g and k positive integers such that −3 ≤ ρ(g, 1, k) < 0. If ρ(g, 1, k) = −3 assume furthermore that k ≥ 6. Then the general k-gonal curve C of genus g has no 𝔤_{d}^{r}’s with ρ(g, r, d) < 0 except 𝔤_{k}^{1} and |K_{C} − 𝔤_{k}^{1}|. In other words the k-gonal locus M_{g,k}^{1} is not contained in any other proper Brill-Noether locus M_{g,d}^{r} with r ≥ 2, d ≤ g − 1 and ρ(g, r, d) < 0.

In seems that other methods are needed to extend this result for more negative values of ρ(g, 1, k).

5 The Kodaira dimension of M_{23}

In this section we explain how Theorem 1 gives a new proof of our result κ(M_{23}) ≥ 2 (cf. [Fa]). We refer to [Fa] for a detailed analysis of the geometry of M_{23} ; in that paper we also conjecture that κ(M_{23}) = 2 and we present evidence for such a possibility.

Let us denote by \overline{M}_{g} the moduli space of Deligne-Mumford stable curves of genus g. We study the multicanonical linear systems on \overline{M}_{23} by exhibiting three explicit multicanonical divisors on \overline{M}_{23} which are (modulo a positive combination of boundary classes coming from \overline{M}_{23} − M_{23}) of Brill-Noether type, that is, loci of curves having a 𝔤_{d}^{r} when ρ(23, r, d) = −1.

On M_{23} there are three Brill-Noether divisors corresponding to the solutions of the equation ρ(23, r, d) = −1: the 12-gonal divisor M_{23,12}^{1}, the divisor M_{23,17}^{2} of curves having a 𝔤_{17}^{2} and finally the divisor M_{23,20}^{3} of curves possessing a 𝔤_{20}^{3}. If we denote by \overline{M}_{g,d}^{r} the closure of M_{g,d}^{r} inside \overline{M}_{g}, the classes [\overline{M}_{g,d}^{r}] ∈ Pic_{ℚ}(\overline{M}_{g}) when ρ(g, r, d) = −1 have been computed (see [EH],[Fa]). It is quite remarkable that for fixed g all classes [\overline{M}_{g,d}^{r}] are proportional. One also knows the canonical divisor class (cf. [HM]):

K_{\overline{M}_{g}} = 13λ − 2δ_{0} − 3δ_{1} − 2δ_{2} − ⋯ − 2δ_{[g/2]},

and by comparing for g = 23 this formula with the expression of the classes [\overline{M}_{23,d}^{r}], we find that there are constants m, m_{1}, m_{2}, m_{3} ∈ ℤ_{>0} such that

mK_{\overline{M}_{23}} = m_{1}[\overline{M}_{23,12}^{1}] + E = m_{2}[\overline{M}_{23,17}^{2}] + E = m_{3}[\overline{M}_{23,20}^{3}] + E,

where E is the same positive combination of the boundary classes δ_{1}, …, δ_{11}. As explained in [Fa], since \overline{M}_{23,12}^{1}, \overline{M}_{23,17}^{2} and \overline{M}_{23,20}^{3} are mutually distinct irreducible

divisors, we can show that the multicanonical image of \overline{M}_{23} cannot be a curve once we construct a smooth curve of genus 23 lying in the support of exactly two of the divisors M_{23,12}^{1},M_{23,17}^{2} and M_{23,20}^{3}. In this way we rule out the possibility of all three intersections of two Brill-Noether divisors being equal to base-locus(|mK_{\overline{M}_{23}}|) ∩ M_{23}.

In [Fa] we found such genus 23 curves using an intricate construction involving limit linear series (cf. Proposition 5.4 in [Fa]). Here we can construct such curves in a much simpler way by applying Theorem 1 when (d, g) = (18, 23): there exists a smooth curve C ⊆ ℙ^{3} of genus 23 and degree 18 such that gon(C) = 13; moreover C sits on a smooth quartic surface S ⊆ ℙ^{3} such that Pic(S) = ℤC ⊕ ℤH.

Since C has a very ample 𝔤_{18}^{3}, by adding 2 base points it will also have plenty of 𝔤_{20}^{3}’s and also 𝔤_{17}^{2}’s of the form 𝔤_{18}^{3}(−p) = {D ∈ 𝔤_{18}^{3} : D ≥ p}, for any p ∈ C. Therefore [C] ∈ (M_{23,20}^{3} ∩ M_{23,17}^{2}) − M_{23,12}^{1}, and Theorem 2 now follows.

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

e-mail: gfarkas@math.lsa.umich.edu