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LINEAR SYZYGIES OF k-GONAL CURVES

GAVRIL FARKAS AND MICHAEL KEMENY

Abstract. In this paper we consider the linear strand of the minimal free resolution of a k-gonal curve C of genus g. We firstly study Schreyer’s Conjecture, which goes beyond Green’sconjecture in predicting finer Brill–Noether type properties of an embedded curve in terms ofthe extremal Betti numbers. In particular, we verify this conjecture for generic k-gonal curves.

Secondly, we consider the vanishing of the Koszul groups Kp,1(C,L) for nonspecial linebundles on a curve C by formulating an effective version of the Green–Lazarsfeld gonalityconjecture and proving it for general k-gonal curves. This generalizes the asymptotic gonalityconjecture proved by Ein–Lazarsfeld [EL] and improves results of J. Rathmann [R] in the casewhere C is general.

0. Introduction

0.1. Minimal free resolutions. Consider a complex algebraic curve C, embedded in projectivespace by a very ample line bundle L. The study of the equations defining the embedded curveC → Pn goes by the name of syzygies.

To be precise, consider the finitely generated, graded Sym H0(C,L) ' C[x0, . . . , xn] moduleΓC(L) :=

⊕nH

0(C, nL). If C → Pn is projectively normal, then ΓC(L) is the homogeneouscoordinate ring OC .

The Hilbert Syzygy Theorem states that we have a minimal free resolution

0→ Fn+1δn+1−−−→ Fn . . .→ F0

δ0−→ ΓC(L)→ 0

of length at most n+ 1, [Ei, Thm. 1.1]. Write Fi in terms of its graded pieces

Fi =⊕j

Vi,j ⊗ S(−j),

where Vi,j is a complex vector space. The Syzygy Spaces are defined as

Ki,j(C,L) := Vi,i+j .

Our task is to determine the Betti numbers bi,j defined by bi,j(C,L) := dimKi,j(C,L). When itis unlikely to cause confusion, we simplify the notation by writing bi,j = bi,j(C,L).

To better interpret the information given by the Betti numbers, consider the Betti table, thatis, the table with (i, j)th entry given by bj,i.

j\i 0 1 2 . . .0 b0,0 b1,0 b2,0 . . .1 b0,1 b1,1 b2,1 . . .2 b0,2 b1,2 b2,2 . . ....

In practice, the two most interesting rows of the Betti diagram are the 1st and 2nd rows. The

1st row, that is, the row with entries bi,1, is called the linear strand. The 2nd row is called thequadratic strand. The relationship between the intrinsic geometry of an embedded curve andthe quadratic strand has been studied extensively in [FK], [K2]. In this paper we will focusprimarily on the linear strand.

1

2 G. FARKAS AND M. KEMENY

0.2. Invariants of the Betti table. There are several natural invariants of an embedded curvewhich one can read directly off the Betti table. The first of these to receive attention are thequantities

l1(C,L) := maxj ∈ N>0 | bj,1(C,L) 6= 0.l2(C,L) := minj ∈ N>0 | bj,2(C,L) 6= 0.

As usual, when the curve and the line bundle are understood, we write li = li(C,L) for i = 1, 2.In the case L = ωC , l2(C,ωC) has been called the linear colength, [SSW]. For example, thecelebrated Green’s conjecture states that the linear colength of a canonically embedded curve isequal to the Clifford index of the curve [G].

To equally important invariants of the Betti table are the extremal Betti numbers

m1(C,L) := bl1,1(C,L)

m2(C,L) := bl2,2(C,L).

0.3. Schreyer’s Conjecture. Consider the case L = ωC of a canonically embedded curve. Aconjecture of Schreyer gives a prediction of the invariant m1 = m1(C,ωC) in terms of the Brill–Noether theory of the curve C, [Sch1, §6], [SSW]. To motivate Schreyer’s Conjecture, we givethree sample Betti tables of curves of genus 9, taken from [S]. Recall that the gonality of a curveis the least value of k, such that C admits a g1

k.

Example 1 ([S]). The following is the Betti diagram of a general curve of genus 9 and gonality5.

j\i 0 1 2 3 4 5 6 71 21 64 70 42 4 70 64 21

Thus m1 = 4.The following is the Betti diagram of a curve of genus 9 and gonality 5 which admits two g1

5’s.

j\i 0 1 2 3 4 5 6 71 21 64 70 82 8 70 64 21

We have m1 = 8.The following is the Betti diagram of a general curve of genus 9 and gonality 4.

j\i 0 1 2 3 4 5 6 71 21 64 75 24 52 5 24 75 64 21

We have m1 = 5.

Experiments such as the above suggests the following definition:

Definition 0.1 ([SSW]). A curve C of genus g and gonality k < bg+32 c is goneric if l2 = k − 2

and m1 = g − k.

Conjecture 0.2 (Schreyer’s Conjecture). Let C be a curve of genus g and gonality 3 ≤ k <

bg+32 c. Assume W 1

k (C) is reduced and irreducible of dimension zero. Assume in addition that

the point [A] ∈ W 1k (C) is the unique line bundle of degree ≤ g − 1 achieving the Clifford index.

Then C is goneric.

One direction of this conjecture is known. Indeed, if the hypotheses of the conjecture aboveare not satisfied, then it is easy to see that C fails to be goneric, [SSW, Prop. 4.10]. Note thebound bg−k,1 ≥ g − k holds for any curve of gonality k, by considering the syzygies of the scrollspanned by the g1

k, see Lemma 4.3.

LINEAR SYZYGIES OF k-GONAL CURVES 3

Schreyer’s Conjecture is known to hold for a general curve of gonality, provided (k− 1)2 < g,[Sch2]. Outside this range, little has been known. We show the following

Theorem 0.3. Assume C is a general smooth curve of genus g and gonality 3 ≤ k < bg+32 c.

Then C is goneric.

As Green’s conjecture is known for general curves in each gonality stratum [V1], [V2], thecondition l2 = k− 2 holds for the general curve of each gonality. Thus, Schreyer’s Conjecture inthe case of generic curves purely concerns the condition bg−k,1 = g − k.

0.4. The Effective Gonality Conjecture. The following conjecture is due to M. Green andR. Lazarsfeld1, [GL1, Pg. 86]

Conjecture 0.4 (Effective Gonality Conjecture). Let L be a globally generated line bundle ofdegree d on a curve C of genus g and gonality k such that d ≥ 2g − 1 + k. Then

l1(C,L) ≤ h0(L)− k − 1.

In other words, the conjecture predicts Ki,1(C,L) = 0 for i ≥ h0(L) − k. The bound on thedegree in the Effective Gonality Conjecture is optimal. Indeed, if E ∈W 1

k , then

Kg−1,1(C,ωC ⊗ E) 6= 0,

by the Green–Lazarsfeld nonvanishing theorem, [G, Appendix]. If k ≤ 3 then this conjecturefollows from [G, Thm. 3.c.1]. Thus we will from now on assume k ≥ 4.

An asymptotic version of the conjecture has recently been proven in [EL]. To date, the bestbound for the degree is due to Jurgen Rathmann who proves that the result holds for d ≥ 4g−3.

Our result is then:

Theorem 0.5. Let C be a general curve of genus g and gonality k. Then the Effective GonalityConjecture holds on C, i.e. for any line bundle of degree at least 2g − 1 + k on C we haveKi,1(C,L) = 0 for i ≥ h0(L)− k.

0.5. Organisation of the Paper and Outline of the Proofs. We first review some back-ground on syzygies in Section 1. In Section 2, we prove Theorem 0.5. We do this by firstconsidering the case of a curve of odd genus g = 2n+1 and maximal gonality n+2. In this case,the divisorial case of the Secant Conjecture (Theorem 1.4 of [FK]) together with arguments onthe dimension on secant varieties (cf. [AS], [FK]) combine to give:

Theorem 0.6. Let C be a smooth curve of genus 2n+ 1 and gonality n+ 2. Then for any linebundle of degree at least 5n+ 3 we have Ki,1(C,L) = 0 for i ≥ h0(L)− n− 2.

Note that Theorem 0.6 is valid for an arbitrary smooth curve of odd genus and maximal gonality.In fact, the proof of Theorem 0.6, also shows that any line bundle of degree 5n + 2 on such acurve with

Kh0(L)−n−2,1(C,L) 6= 0

must be of the form L = KC + E for E ∈W 1n+2(C), see Proposition 2.1.

In order to deduce Theorem 0.5, we fix a value for the gonality k and perform induction on thegenus g ≥ 2k−3; the initial step is Theorem 0.6. By induction, assume that the general smoothcurve C of genus g and gonality k satisfies the Effective Gonality Conjecture. We then constructa stable curve X of genus g + 1 by adding an elliptic curve E at a point of ramification of theg1k; then X is a limit of curves of gonality k by Chapter 3.G [HM1]. An analysis of syzygies of

1In fact, Green and Lazarsfeld actually only stated their conjecture above for deg(L) >> 0, and merely raisedthe above effective version as a possibility.


line bundles of bidegree (2g + k, 1) on X allows us to deduce the Effective Gonality Conjecturefor small deformations of X.

We prove Theorem 0.3 in Sections 3, 4 and 5. We begin in Section 3 with the observation thatthe main result of [HR] implies that a smooth curve C of genus 2k − 1 and gonality k satisfiesbk−1,1(C) = k − 1, provided W 1

k (C) is integral of dimension zero. Now consider the Hurwitzspace Hk of smooth curves which are degree k covers of P1. We have two divisors on Hk: thedivisor Syz parametrizing points with bk−1,1 > k − 1, (considered in [B]), and the divisor BNparametrizing points which have an extra g1

k, [AC]. By the above discussion these two divisorscoincide set-theoretically.

Now suppose we are no longer in the divisorial case. Broadly speaking, we follow the strategyemployed in [Ap2], but in our situation there are serious new obstacles. Assuming that Csatisfies a generality condition, after choosing g + 1 − 2k general pairs of points, we producea point [D] ∈ Hg+1−k. Assuming in addition that W 1

k (C) is integral, then one checks directlythat, set-theoretically W 1

g+1−k(D) consists of one point, namely the torsion free sheaf given by

pushing forward the unique g1k on C.

Unfortunately, the scheme structure of W 1g+1−k(D) is rather difficult to analyse and seems

likely to be non-reduced, which is the main technical difficulty we face. Nevertheless, using ideasrelated to limit linear series, we eventually show that [D] /∈ BN, provided that D is suitablygeneral. To conclude bk−1,1 = k−1, in Section 4 we extend the construction of the syzygy divisorto an open set containing [D] (in an appropriate moduli space). In the short Section 5 we useK3 surfaces to show that the extended Koszul divisor does not contain the unique boundarycomponent ∆ containing [D], completing the proof.

Acknowledgments: We thank C. Bopp, M. Hoff and F.-O. Schreyer for stimulating conversa-tions on Schreyer’s Conjecture. This work was supported by the DFG Priority Program 1489Algorithmische Methoden in Algebra, Geometrie und Zahlentheorie.

1. Background on Syzygies

We firstly recall two results which, taken together, reduce Conjecture 0.4 to the case d =2g − 1 + k, i = h0(L)− k = g. The following is due to Marian Aprodu.

Theorem 1.1 ([Ap], Lemma 4.1). Let C be a smooth curve of genus g and L a line bundle ofdegree d ≥ g with h1(L) = 0. Assume Kp,1(C,L) = 0. Then Kp+1,1(C,L+ x) = 0 for any pointx ∈ C such that L+ x is base point free.

The following lemma is rather standard, see [AN], Corollary 2.13.

Lemma 1.2. Let C be a smooth curve and let L be a globally generated line bundle with L OC .Suppose Kp,1(C,L) = 0. Then Kp+1,1(C,L) = 0.

Let X be an arbitrary projective variety, possibly singular, and let L,M ∈ Pic(X) be arbitraryline bundles, not necessarily base point free. Consider the graded Sym H0(X,L) module

ΓX(M,L) :=⊕n∈Z

H0(X,nL+M).

One may define the Koszul groups Kp,q(X,M ;L) using the Koszul complex, [G].The following lemma is very well-known:

Lemma 1.3 (Semicontinuity). Let π : X → S be a flat, projective morphism of Noetherianschemes and assume S is integral. Let L ∈ Pic(X ) be a line bundle, and assume that

h0(Xs,Ls) = c


is independent of s ∈ |S|. Let M∈ Pic(X ) be a second line bundle, and assume

h0(Xs, (q − 1)Ls ⊗Ms) = r1, h0(Xs, qLs ⊗Ms) = r2, h

0(Xs, (q + 1)Ls ⊗Ms) = r3

are also independent of s ∈ |S|. Then the function

ψ : |S| → Zs 7→ dimKp,q(Xs,Ms;Ls)

is upper semicontinuous.

Proof. By Grauert’s Theorem E := π∗L, F1 := π∗(L⊗q−1 ⊗M), F2 := π∗(L⊗q ⊗M), F3 :=π∗(L⊗q+1⊗M) are all vector bundles. It is clearly enough to assume that S = Spec(R) is affine.We have a Koszul complex

p+1∧E ⊗ F1

δ1−→p∧E ⊗ F2

δ2−→p−1∧E ⊗ F3,

where both maps are R module morphisms. For any p ∈ Spec(R), Kp,q(Xp,Mp;Lp) is given bythe middle cohomology of

(

p+1∧E ⊗ F1)⊗ k(p)

δ1⊗k(p)−−−−−→ (

p∧E ⊗ F2)⊗ k(p)

δ2⊗k(p)−−−−−→ (

p−1∧E ⊗ F3)⊗ k(p).

Hence

dimKp,q(Xp,Mp;Lp) = dim Ker(δ2 ⊗ k(p))− dim Im(δ1 ⊗ k(p))

= r2

(c

p

)− dim Im(δ2 ⊗ k(p))− dim Im(δ1 ⊗ k(p)).

So it suffices to show that, for any morphism ψ : A→ B of finitely generated, free R modules,the function p 7→ rank ψ ⊗ k(p) is lower semicontinuous. But for any r ∈ N the set

p ∈ Spec(R) | rank(ψ ⊗ k(p)) < ris closed, with ideal given by the entries of the matrix ∧rψ.

2. The Effective Gonality Conjecture for generic curves

We start by proving Theorem 0.6.

Proof of Theorem 0.6. Let C be a curve of genus 2n+ 1 and gonality n+ 2. We need to provethat, for any line bundle of degree at least 5n+ 3 we have Ki,1(C,L) = 0 for i ≥ h0(L)− n− 2.As explained in the introduction, we may assume n ≥ 2. By Lemmas 1.1 and 1.2, it is enoughto prove that, for any line bundle L of degree exactly 5n+ 3 we have K2n+1,1(C,L) = 0.

Theorem 1.4 of [FK] states that, for any line bundle M of degree 4n+ 2

Kn−1,2(C,M) 6= 0⇔M −KC ∈ Cn+1 − Cn−1,

since a curve of odd genus is of maximal gonality if and only if it is of maximal Clifford index,[HR, Remark 6.3]. For any line bundle M ∈ Pic4n+2(C)

dimKn,1(C,M)− dimKn−1,2(C,M) = 0,

by [AF, Theorem 2.8]. Thus, for any M ∈ Pic4n+2(C),

Kn,1(C,M) 6= 0⇔M −KC ∈ Cn+1 − Cn−1.

Using Lemma 1.1 again, it thus suffices to show that, for any line bundle L of degree 5n + 3,there exists a divisor D ∈ Cn+1 such that

L−D −KC ∈ Cn+1 − Cn−1.


Suppose this were not the case, i.e. suppose that

L−KC − Cn+1 ⊆ Cn+1 − Cn−1.

Then for every D ∈ Cn+1 there exists some E ∈ Cn+1 such that H1(C,L − (D + E)) 6= 0, i.e.D + E is an element of the secant variety V 2n+1

2n+2 (L). In particular,

dimV 2n+12n+2 (L) ≥ n+ 1,

which is one higher than the expected dimension n. We have a morphism

ψ : V 2n+12n+2 (L)→ Cn−1

A 7→ KC − L+A.

Let I be any component of V 2n+12n+2 (L) of dimension n+ 1 and r := n− 1− dim(ψ(I)). Then ψ|I

must have fibres of dimension at least 2 + r. As all divisors in the inverse image ψ−1(B) areclearly linearly equivalent, we have h0(A) ≥ 3 + r for all A ∈ V 2n+1

2n+2 (L) such that ψ(A) ∈ ψ(I).

By Riemann–Roch, this implies h1(A) ≥ 1 + r, or h0(KC − A) = h0(2KC − L−B) ≥ 1 + r forall such A. The latter holds for any effective divisor B ∈ ψ(I), so we must have dim |2KC −L| ≥ r + dimψ(I) = n − 1. This is equivalent to h1(2KC − L) ≥ 3, which is equivalent toL −KC ∈ W 2

n+3. But this implies Cliff(C) ≤ n − 1 (if n = 2, then compute the Clifford indexof 2KC − L rather than L−KC). Since we have Cliff(C) = n, this is a contradiction.

In fact, the proof of the previous theorem gives a characterisation of those line bundles L ofdegree 2g− 2 + k such that the group Kh0(L)−k,1(C,L) is nonvanishing, in the case where C hasodd genus and maximal gonality.

Proposition 2.1. Let C be a smooth curve of odd genus 2n + 1 and gonality n + 2. LetL ∈ Pic5n+2(C) be such that K2n,1(C,L) 6= 0. Then L−KC ∈W 1

n+2(C).

Proof. Following the proof of Theorem 0.6, we obtain dimV 2n2n+1(L) ≥ n. By studying the

morphism

ψ : V 2n2n+1(L)→ Cn−1

A 7→ KC − L+A.

and arguing as in Theorem 0.6, we are again led to the statement h0(2KC −L) ≥ n. Riemann–Roch now gives h0(L−KC) ≥ 2, as required.

We will prove Theorem 0.5 by an induction on the genus, fixing the gonality. To perform theinduction step, let C be a smooth, genus g curve of gonality k and assume in addition that thereexists a g1

k such that the induced map C → P1 is simply branched. Let p ∈ C be a branch pointof C → P1, and consider the stable curve X = C ∪p E obtained by glueing a smooth, genus 1curve at p. X is a limit of smooth, genus g + 1 curves of gonality k, [HM2, §3.G].

Proposition 2.2. Let X = C ∪p E be the genus g + 1 stable curve as above. Let L be a linebundle on X such that deg(L|C ) = 2g+k and deg(L|E ) = 1. Then, for a general point q ∈ E \p,

Kg,1(X,L(−q)) = 0.

Further, for such a point h1(X,L(−q)) = h1(X, 2L(−2q)) = 0.

Proof. We have the Mayer–Vietoris sequence

0→ L|C (−p)→ L(−q)→ L|E (−q)→ 0.


The degree of L|E (−q) = 0, so, if q ∈ E\p is general, we have h0(E, jL(−jq)) = h1(E, jL(−jq)) =

0 for j = 1, 2, which implies h1(X,L(−q)) = h1(X, 2L(−2q)) = 0. Further, we have a naturalisomorphism H0(C,L(−p)) ' H0(X,L(−q)), and we know, by the assumptions on C, that

Kg,1(C,L(−p)) = 0.

We will use this to deduce Kg,1(X,L(−q)) = 0.We have a natural commutative diagram:∧g+1H0(C,L(−p)) d−−−−→

∧gH0(C,L(−p))⊗H0(C,L(−p)) d−−−−→∧g−1H0(C,L(−p))⊗H0(C, 2L(−2p))

α

y β

y γ

y∧g+1H0(X,L(−q)) d−−−−→∧gH0(X,L(−q))⊗H0(C,L(−q)) d−−−−→

∧g−1H0(X,L(−q))⊗H0(X, 2L(−2q)),

where α, β are isomorphisms, and γ is induced from the natural composition

H0(C, 2L(−2p)) → H0(C, 2L(−p)) ' H0(X, 2L(−2q)).

As Kg,1(C,L(−p)) = 0 the top row is exact, and since β is surjective and γ is injective, thebottom row must also be exact, as required.

From Proposition 2.2 we readily deduce Theorem 0.5.

Proof. Fix k ≥ 4. Assume that the general curve C of genus g has the property that, for anyline bundle L ∈ Pic2g−1+k(C), Kg,1(C,L) = 0. We claim there exists a smooth curve C ′ of genus

g + 1 such that, for each line bundle L′ ∈ Pic2g+1+k(C ′), Kg+1,1(C ′, L′) = 0. By performinginduction on k and noting that the initial step is Theorem 0.6, this suffices to prove the theorem.By Theorem 1.1 it further suffices to prove that there exists a smooth curve C ′ of genus g + 1and gonality k such that, for each line bundle L′ ∈ Pic2g+1+k(C ′), there exists a point q ∈ C ′such that Kg,1(C ′, L′(−q)) = 0.

Let X = C ∪p E be the genus g + 1 stable curve as in Proposition 2.2. Consider a flat familyπ : C → S of stable curves over a smooth, pointed, one dimensional base (S, 0), such that thecentral fibre is X and π−1(s) is a smooth curve of gonality k for all 0 6= s ∈ S; by the theoryof admissible covers such families exist. As X is a curve of compact type and after shrinking Sand performing a finite base change if necessary, we have a relative Picard scheme

f : Pic2g+1+k(C/S)→ S,

with central fibre consisting of all line bundles of multidegree (2g + k, 1) on X = C ∪p E; thisscheme is flat and proper over S, see [D, §4] and [EH], proof of Thm. 3.3.

Let C0 be the open set C \p of all points which are smooth in the fibres over S. By Proposition2.2 together with semicontinuity for the dimension of Koszul groups, there is an open subsetU ⊆ Pic2g+1+k(C/S) ×S C0, such that, for each pair (L′, q′) ∈ U , Kg,1(C ′, L′(−q)) = 0, whereC ′ = π−1(f(L′)), and such that

0 /∈ f(Pic2g+1+k(C/S) \ pr1(U)),

where pr1 : Pic2g+1+k(C/S)×S C0 → Pic2g+1+k(C/S) is the projection. As flat morphisms areopen, pr1(U) is open, and since f is proper, V := f(Pic2g+1+k(C/S) \ pr1(U)) is closed. Thusif 0 6= t ∈ S \ V and Ct := π−1(t), then, for each L ∈ Pic2g+1+k(Ct) there exists q ∈ Ct withKg,1(Ct, L(−q)) = 0, as required.


3. Schreyer’s Conjecture for general curves

Let X be a complex manifold and let M(x) = (ai,j) be an n × n matrix, with entries holo-morphic functions ai,j(x) : X → C.

Lemma 3.1. Let M(x) be a matrix as above and assume that M(p) has rank ≤ r for somep ∈ X. Then the determinant det(M(x)) vanishes to order ≥ n− r at p.

Proof. We may clearly assume that the scalar matrix M(p) is in Jordan normal form. The claimthen follows eaily by induction on n, the case n = 1 being obvious, and is left to the reader.

Set g = 2k − 1 and consider the following two loci on Mg

Syz : = [C] ∈Mg | Kk−1,1(C,ωC) 6= 0Hur : = [C] ∈Mg | C carries a g1

d.

Recall Syz and Hur are defined as degeneracy loci and their closures define divisors on Mg,[HR]. Further, we have the linear equivalence

Syz = (k − 1)Hur.

Theorem 3.2 ([HR]). Suppose [C] ∈ Mg is a smooth curve of gonality k such that the point[C] ∈ Hur is smooth. Then bk−1,1(C) = k − 1.

Proof. From [HR], there exist two vector bundles V and W of the same rank over Mg, witha morphism t : V → W , such that, for any [C] ∈ Mg, we may identify Kk−1,1(C,ωC) withKer(t ⊗ [C]). Then Syz is defined by det(t). Suppose bk−1,1(C) ≥ k. Then Lemma 3.1 states

that det(t) vanishes to order at least k. By the equation Syz = (k−1)Hur, the function definingHur must vanish to order at least two. Thus Hur is not smooth at [C].

Corollary 3.3. Let C be a smooth curve of genus g = 2k − 1 and gonality k. Assume W 1k (C)

is reduced and irreducible. Then bk−1,1(C) = k − 1.

Proof. Consider G1k → Mg, the universal G1

k, [ACGHII, Ch. XXI]. The space is smooth and,locally over [C], it is isomorphic to π : W1

k → Mg. By our assumption, π is unramified nearthe unique point [A] ∈W 1

k (C), [ACGHII], §12, XXI. Thus Hur is smooth at [C].

We now turn our attention to curves of genus g and arbitrary gonality k < bg+32 c. We begin

by recalling the following, see [ACGHII], §12, XXI, [AC, §1].

Theorem 3.4. Let C be a curve of genus g and gonality k < bg+32 c. Let L ∈ W 1

k (C) be base

point free with h0(C,L) = 2. Then W 1k (C) is reduced of dimension zero at [L] if and only

h0(C,L2) = 3.

Let C be a smooth curve and W 1k (C) the locus of line bundles A with degree k and at least

two sections: W 1k (C) comes with the usual determinantal scheme structure. We let W 1,bpf

k (C) ⊆W 1k (C) denote the open locus of base point free line bundles. We make the following definition,

which is a slight modification of the linear growth condition of [Ap2]:

Definition 3.5. A curve C of gonality d is said to satisfy “bpf-linear growth” if

dimW 1d+m(C) ≤ m for 0 ≤ m ≤ g + 1− 2d

and, further, dimW 1,bpfd+m (C) < m for 0 < m ≤ g + 1− 2d.

In the above definition, we make the convention dim ∅ = −∞.


Lemma 3.6. Let C be a general curve of genus g and gonality 2 ≤ k < bg+32 c. Then C satisfies

bpf-linear growth.

Proof. We need to show dimW 1d+k(C) ≤ m for 0 ≤ m ≤ g+1−2k, and, in addition dimW 1,bpf

k+m (C) <

m, 0 < m ≤ g + 1 − 2k. We firstly deal with the condition dimW 1d+k(C) ≤ m for 0 ≤ m ≤

g+ 1− 2k. For k = 2, this follows from [M], so we assume k ≥ 3. By [Ap2], C satisfies ordinarylinear growth for k ≥ 3, i.e. dimW 1

d+k(C) ≤ m for 0 ≤ m ≤ g + 2− 2k, so the first condition iscertainly satisfied.

For the second condition, we follow [Ap2], proof of Theorem 2, and let S →Mg be an etaleatlas, let K be an irreducible component of G1

k ×S G1k+m which dominates G1

k , for 0 < m ≤g+ 1− 2k. We assume that a general point of K is a pair (A,B), with A, B base point free linebundles on a smooth curve C with A a simple, minimal pencil, and wish to prove

dimK ≤ dimG1k +m− 1 = 2g + 2k − 6 +m.

Firstly suppose that if (A,B) ∈ K is an general point, then (A,B) is not composed with aninvolution. Then [AC, Prop. 2.4] states that dimK = g + 2k+ 2(k+m)− 7 ≤ 2g + 2k− 6 +m,since m ≤ g+ 1− 2k. Now suppose that a general point (A,B) is composed with an involution,i.e. the morphism ψ = (φA, φB) : C → P1 × P1 is not birational to its image. Consider the firstprojection pr1 : ψ(C) → P1. As φA : C → P1 has simple ramification, pr1 must have degreeone, and hence must be an isomorphism.

For C smooth, let

Grd(C) := (L, V ) : L ∈ Picd(C), V ∈ G(r + 1, H0(C,L))

be the variety parametrising grd′s, [ACGH, §IV.3]. Let Gr,bpfd (C) ⊆ Grd(C) be the open subset

of pairs (L, V ) such that V is base point free. The next lemma states that one can characterizebpf-linear growth in terms of grd

′s.

Lemma 3.7. A curve C of genus g and gonality d satisfies bpf-linear growth if and only if

dimG1d+m(C) ≤ m for 0 ≤ m ≤ g + 1− 2d

and, further, dimG1,bpfd+m (C) < m for 0 < m ≤ g + 1− 2d.

Proof. We have natural surjections G1k(C)→W 1

k (C) and G1,bpfk (C)→W 1,bpf

k (C), so, if C satis-

fies dimG1d+m(C) ≤ m for 0 ≤ m ≤ g + 1− 2d and dimG1,bpf

d+m (C) < m for 0 < m ≤ g + 1− 2d,then it satifies bpf-linear growth.

For the other direction, observe that if I ⊆ W rk (C) is an irreducible component, then I ∩

W r+1k (C) has codimension at least two in I, provided g − k + r ≥ 0. This follows from the fact

that no component of Crk is entirely contained in Cr+1k under the above bound, [ACGH, §IV.1],

by considering fibres of the natural morphism Crk → W rk (C). So suppose C satisfies bpf-linear

growth. We firstly claim dimG1d+m(C) ≤ m for 0 ≤ m ≤ g + 1− 2d. Take any component

J ⊆ G1d+m(C) and consider the morphism ψ : J → W 1

d+m(C). Suppose the general fibre of ψ

has dimension l ≥ 0. As dimG(2, n) = 2(n − 2), l = 2j must be even, and ψ(J) ⊆ W 1+jd+m(C).

By linear growth and the above, we have dimψ(J) ≤ m − l so that dim(J) ≤ m. Thus

G1d+m(C) ≤ m. By an identical argument dimG1,bpf

d+m (C) < m for 0 < m ≤ g + 1− 2d.

Let D be an integral nodal curve. We define W 1k (D) as the closed subset of the compactified

Jacobian given by rank one, torsion free sheaves A of degree k with h0(D,A) ≥ 2. We definethe gonality of D to be the minimum d such that W 1

d (D) 6= 0.


Proposition 3.8. Let C be a smooth curve of gonality 2 ≤ k < bg+32 c and genus g. Assume C

satisfies bpf-linear growth and, in addition, there exists a unique point [B] ∈W 1k (C). Let (xi, yi),

1 ≤ i ≤ g + 1 − 2k be general pairs of distinct points, and let D be the nodal curve obtainedby glueing xi to yi for all i. Then the set W 1

g+1−k(D) consists of the unique point A = µ∗(B),where µ : C → D is the normalisation morphism. Furthermore D has gonality g + 1− k.

Proof. We follow [Ap2], proof of Theorem 2. Let A ∈ W 1g+1−k(D) and let µ : C → D be the

partial normalisation at nodes of D for which A is singular. Suppose C has arithmetic genus

2g+ 1− 2k−n. Then there is a unique line bundle L ∈ Pic(C) of degree g+ 1− k−n such that

µ∗(L) = A. We need to prove n = g + 1− 2k, or, equivalently, that C is smooth.

Now C has m = g + 1 − 2k − n nodes. Suppose for a contradiction that m > 0 and let

ν : C → C be the normalisation. Assume that the nodes lie over (x1, y1), . . . , (xm, ym). If L

has more than two sections, then, by subtracting general smooth points of C and taking thebase point free part, we obtain a subline bundle L′ → L which is base point free, with exactlytwo sections and degree d′ ≤ g + 1 − k − n = m + k. Notice that, by base point freeness, for

each node pi of C, there exists a nonzero section of L′ vanishing at pi. Thus the g1d′ given by

ν∗H0(C, L′) ⊆ H0(C, ν∗L′) ∈ G1,bpfd′(C) has the property that, for each 1 ≤ i ≤ m, there exists

si ∈ ν∗H0(C, L′) with si vanishing at both xi and yi. As the points xi, yi are general, for such

a g1d′ to exist, we must have dimG1,bpf

d′ (C) ≥ m, which contradicts the assumption of bpf-lineargrowth and Lemma 3.7.

The proof that D has gonality g + 1 − k is essentially the same as the argument in [Ap2],proof of Theorem 2, and is omitted.

Consider the moduli space Admg,k of degree k admissible covers of genus g, [HM2], [ACV].We have a morphism

π : Admg,k →Mg

given by sending [f : B → T ] to the stabilization of B. The image of π is Hur.The following is the translation of Proposition 3.8 to the moduli space of admissible covers.

Proposition 3.9. Let C be a smooth curve of gonality 2 ≤ k < bg+32 c and genus g. Assume

C satisfies bpf-linear growth and that there exists a unique [A] ∈ W 1k (C). Let (xi, yi), 1 ≤ i ≤

g+ 1− 2k be general pairs of distinct points, and let D be the nodal curve obtained by glueing xito yi for all i. Assume in addition that all ramification of the morphism φA : C → P1 is simple.Set n = g + 1− 2k and consider

π : Admg+n,k+n →Mg+n.

Then the underlying set of π−1([D]) consists of a unique point.

Proof. We will show that the construction in [HM2, Thm. 5] is unique in our case. Let f : B → T

be an admissible cover of degree k+n and suppose the stabilisation B is isomorphic to D. Thereexists a unique component C0 of B such that C0 has genus > 0 and C0 is a partial normalisationof D. The restriction f0 := f|C0

gives a morphism f0 : C0 → P10 to a copy of P1. By admissibility,

C0 ' C and deg(f) ≥ k.We now show f0 cannot identify any pair xi, yi. Assume that f0 identifies the pairs xi, yi for

1 ≤ i ≤ j. As the stabilisation of B is D, for each pair xi, yi, j + 1 ≤ i ≤ n there must be

a component Ri ∈ B \ C0 containing xi and yi. Then Ri must have genus 0, and furthermore

f(Ri) gives a path from f(xi) to f(yi) ∈ T . As T has genus 0, f(Ri) must contain P10 (or else

T contains a loop). Thus Ri contains a component Ci mapping with degree 1 to P10 for each

i ≥ j + 1. As deg(f) = n+ k, this implies that deg(f0) ≤ n+ k − j.


Figure 1. An admissible cover of degree 4.

Let C be the nodal curve obtained by identifying xi, yi for 1 ≤ i ≤ j. Then f0 factors through

a map g : C → P1. But then, there is a base point free L ∈ Pic(C) with at least two sectionsand deg(L) ≤ n+ k − j. Pushing forward to D, we obtain a torsion free sheaf A′ ∈ W 1

d (D) ford ≤ n+ k, nonsingular at the nodes of xi, yi for 1 ≤ i ≤ j. This contradicts Proposition 3.8.

This implies that the components Ci constructed above are unique and defined for all 1 ≤ i ≤ nso that deg(f0) = k. Further, by the assumptions on C, the morphism f0 must be φA : C → P1.

As φA has only simple ramification and the stability of the marked curve T , it follows thatB has no tails at the ramification points of φA. We must mark T at all 2g − 2 + 2k points over

the ramification points, leaving 4n markings left. Now, consider the component Rxi of Ri \ Cicontaining xi. As T has no loops, this meets Ci at a unique point mapping to xi. Likewise,we define Ryi . Notice that all f(Rxi), f(Ryi) are pairwise disjoint and meet P1

0 uniquely atpoints f(xi) resp. f(yi). In order for T to be stable, we claim that there must be at least 2marked points on each f(Rxi), f(Ryi). This is clear if f(Rxi) resp. f(Ryi) has one component.Otherwise, f(Rxi) resp. f(Ryi) has at least two extremal components2, at most one of whichmeets P1

0, so f(Rxi) resp. f(Ryi) needs at least 3 marked points to be stable. Since we onlyhave 4n marked points left, we conclude that f(Rxi), f(Ryi) are integral and contain exactlytwo marked points. This implies that the Rxi , Ryi are themselves integral (using admissibility)and that f|Rxi

, resp. f|Rxihas degree 2.

The morphism f is then uniquely made into an admissible cover by adding tails mappedwith degree 1 to the f(Rxi) resp. f(Ryi) at all remaining points where this is possible withoutbreaking admissibility.

The unique admissible cover in π−1([D]) is drawn above for k = 2, n = 2.

2A component is extremal if the corresponding vertex in the dual graph has degree 1. When the dual graph isa tree with at least two vertices, at least two such components exist by the degree-sum formula.


From now on set n = g + 1− 2k and assume n ≥ 1. Let

BN ⊆ Admg+n,k+n ×Mg+nAdmg+n,k+n

be the closure of the locus of pairs (f : C → P1, g : C → P1) with C smooth and f g. By

[AC, Prop. 2.4], dim BN = dimAdmg+n,k+n − 1. Set

BN := pr1(BN) ⊆ Admg+n,k+n.

Each component of BN has dimension at most dimAdmg+n,k+n − 1. Recall:

Theorem 3.10 ([C]). Let C be a smooth curve of gonality k + n and genus g + n and let[f : C → P1] ∈ Admg+n,k+n. Then h0(C, f∗(O(2))) ≥ 4 implies that [f ] ∈ BN.

Our immediate goal is to show that, in the notation of Proposition 3.9, the point [f ] ∈ π−1([D])is not in BN, provided the normalisation C is sufficiently general. This will be achieved bydegenerating to a nodal curve with k− 2 nodes and such that the normalisation is hyperelliptic.

Proposition 3.11. Let C be a smooth hyperelliptic curve of genus g + 2 − k together with abase point free A ∈ W 1

2 (C). Choose general points y1 and xi on C for 1 ≤ i ≤ g − 1 − k. For2 ≤ i ≤ g − 1 − k, we define yi to be the hyperelliptic conjugate of xi−1 (i.e. yi + xi−1 ∈ |A|).Consider the semistable curve D obtained by adjoining a smooth rational curve Ri to C at xi, yifor each 1 ≤ i ≤ g − 1− k. Let L ∈ Pic(D) be any line bundle such that

• L|C ' A2 +

∑g−1−ki=1 (xi + yi)

• L|Ri' ORi for all 1 ≤ i ≤ g − 1− k.

Then h0(D,L) = 3.

Figure 2. The curve D

Proof. For any power n ≤ g(C)− 1 = g+ 1− k, h0(C,An) = n+ 1. In particular, h0(C,A2) = 3so Riemann–Roch gives h0(C,ωC − A2) = h1(C,A2) = g − k. As y1, x1, . . . , xg−1−k are chosen

generally, h0(C,ωC −A2 − y1 −∑g−1−k

i=1 xi) = 0 which implies h0(C,A2 + y1 +∑g−1−k

i=1 xi) = 3.For each 1 ≤ i ≤ g − 1− k, define

Di := C⋃

1≤j≤iRi.

Define Li ∈ Pic(Di) by Li := L|Di−∑g−1−k

j=i+1 (xj + yj). We will prove by induction on i that

h0(Di, Li) = 3. The inequality h0(Di, Li) ≥ 3 follows from the short exact sequence 0→ A2 →Li →

⊕ij=1ORj → 0, so we need to show h0(Di, Li) ≤ 3.

When i = 1, we have the short exact sequence 0→ OR1(−2)→ L1 → A+x1 + y1 → 0, which

gives h0(D1, L1) ≤ h0(C,A2 + x1 + y1) ≤ h0(C,A2 + y1 +∑g−1−k

j=1 xj) = 3, so the claim holds.


We now prove the induction step. Assume the claim holds for i = j. We claim firstly thateach section of Lj + xj+1 vanishes at xj+1, from which is follows that h0(Dj , Lj + xj+1) =h0(Dj , Lj) = 3, by the induction hypothesis. We have the short exact sequence

0→j⊕l=1

ORl(−2)→ Lj + xj+1 → A2 + xj+1 +

j∑l=1

xl + yl → 0.

So restriction to C gives an injective map H0(Dj , Lj+xj+1) → H0(C,A2 +xj+1 +∑j

l=1(xl+yl))

and it suffices to show the sections of H0(C,A2 +xj+1 +∑j

l=1(xl+yl)) vanish on xj+1. We have

H0(C,A2 + xj+1 +

j∑l=1

(xl + yl)) = H0(C,A2 + y1 + (x1 + y2) + . . .+ (xj−1 + yj) + xj + xj+1)

= H0(C,Aj+1 + y1 + xj + xj+1).

So h0(C,A2 + xj+1 +∑j

l=1(xl + yl)) ≥ h0(C,Aj+1) = j + 2. On the other hand

H0(C,A2 + xj+1 +

j∑l=1

(xl + yl)) ⊆ H0(C,A2 + y1 +

g−1−k∑l=1

xl +

j∑l=2

yl)

and h0(C,A2 + y1 +∑g−1−k

l=1 xl) = 3, so h0(C,A2 + xj+1 +∑j

l=1(xl + yl)) ≤ 3 + j − 1 = j + 2.

Thus H0(C,Aj+1) ' H0(C,A2 + xj+1 +

j∑l=1

(xl + yl)) ' H0(C,Aj+1 + y1 + xj + xj+1). In

particular, each section of H0(C,A2 + xj+1 +∑j

l=1(xl + yl)) vanishes at xj+1 as required, soh0(Dj , Lj + xj+1) = 3.

We now have two cases. In the first case h0(Dj , Lj + xj+1 + yj+1) = 3. Then from

0→ ORj+1(−2)→ Lj+1 → Lj + xj+1 + yj+1 → 0

we see h0(Dj+1, Lj+1) ≤ 3 as required. In the second case, h0(Dj , Lj +xj+1 + yj+1) = 4. In thiscase, first note that, since h0(Dj , Lj + xj+1) = 3, there is a section

t ∈ H0(Dj , Lj + xj+1 + yj+1)

which does not vanish at yj+1.On the other hand, we claim that each section of Lj +xj+1 +yj+1 vanishes at xj+1. As above,

it suffices to show that each global section of A2 +∑j+1

l=1 (xl + yl) vanishes at xj+1. We have

A2+∑j+1

l=1 (xl+yl) = A2+j+y1+xj+1 so that h0(C,A2+∑j+1

l=1 (xl+yl)) ≥ h0(C,A2+j) = j+3 (note

that j+2 ≤ g+1−k). We also have A2 +∑j+1

l=1 (xl+yl) ⊆ A2 +y1 +∑g−1−k

l=1 xl+(y2 + . . .+yj+1)

which gives h0(C,A2 +∑j+1

l=1 (xl + yl)) ≤ 3 + j. Thus

H0(C,A2 +

j+1∑l=1

(xl + yl)) ' H0(C,A2+j + y1 + xj+1) ' H0(C,A2+j)

and each section of A2 +∑j+1

l=1 (xl + yl) vanishes at xj+1.We now conclude the proof from the Mayer-Vietoris sequence

0→ Lj+1 → Lj + xj+1 + yj+1 ⊕ORj+1

α−→ Oxj+1 ⊕Oyj+1 → 0.

Indeed, it suffices to show H0(α) is surjective. By considering the image of t, where t ∈H0(Dj , Lj + xj+1 + yj+1) is the section above, we see that Im(H0(α)) contains the element(0, 1). Next, by considering the images of constants c ∈ H0(ORj+1), we see the image contains

elements (a, b) with a 6= 0. Thus H0(α) is surjective.


Remark 3.12. Suppose we replace the smooth rational curves Ri with a connected, nodal chainof smooth rational curves

Γi := Γi1 ∪ . . .Γikwhere Γij are the components of Γi for 1 ≤ j ≤ k, and with Γi1 meet C at xi (and nowhere else),

Γik meeting C at yi, and no other components meeting C. By a chain, we mean that each Γijmeets Γij+1 and Γij−1 transversally in precisely one point for 2 ≤ j ≤ k − 1, and there are no

other intersections. Then the conclusion (and proof) of Proposition 3.11 holds with Γi replacingRi.

Definition 3.13. Set n = g+1−2k as above. The Koszul divisor Kos ⊆ Admg+n,k+n is definedas the closure of the locally closed set of [f : C → P1] such that dimKg−k,1(C,ωC) > g − k.

From Corollary 3.3, Theorem 3.10 and [SSW, Prop. 4.10] we have the equality of sets

BN = Kos.

We now come to the main result of this section.

Theorem 3.14. Let C be a general smooth curve of gonality k < bg+32 c and genus g. Let

(xi, yi), 1 ≤ i ≤ n = g + 1− 2k be general pairs of distinct points, and let D be the nodal curveobtained by glueing xi to yi for all i. Let the admissible cover [f : B → T ] be the unique elementof π−1([D]) as in Proposition 3.9. Then [f ] /∈ Kos.

Proof. It suffices to show [f ] /∈ BN. As the underlying set of π−1([D]) is a point, it further

suffices to show [f, f ] /∈ BN ⊆ Admg+n,k+n ×MgAdmg+n,k+n. To prove this, specialise C to a

nodal curve C0 with k − 2 nodes, such that the normalization C0 of C0 is hyperelliptic. Note

that, if A ∈W 12 (C0) and if ν : C0 → C0 is the normalization, then ν∗A ∈W 1

k (C0).

Let (xi, yi), 1 ≤ i ≤ n = g+1−2k be distinct points on C0 and let (xn+j , yn+j), 1 ≤ j ≤ k−2be the points over the node. Let D0 be the curve obtained by glueing xi to yi for 1 ≤ i ≤ g−1−kand let µ : C0 → D0 be the normalisation. Then if A ∈ W 1

2 (C0), µ∗(A) ∈ W 1g+1−k(D0) and D0

is a specialization of D. Assuming (xi, yi) are chosen generally, Proposition 3.9 shows there isa unique admissible cover f0 : B0 → T0 of degree g + 1− k where the stabilization of B0 is D0.

As f0 is a specialisation of f , it suffices to show [f0, f0] /∈ BN.Further specialize by bringing the points (xi, yi), 1 ≤ i ≤ g−1−k into the configuration from

the hypothesis of Proposition 3.11. This specializes f0 to an admissible cover g0 : B′ → T ′, where

the stabilisation of B′ is a nodal curve D′0 with normalisation C0 and nodes in our configuration.

It suffices to show [g0, g0] /∈ BNWe record two properties of the admissible cover g0 : B′ → T ′ which follow from its description

as a limit of the f0 and considerations as in Proposition 3.9. Firstly, B′ has a unique component

of nonzero genus, which isomorphic to the smooth, hyperelliptic curve C0. The restriction of g0

to C0 is the degree two cover of P10 := g0(C0) ' P1, determined by A ∈ W 1

2 (C0). Secondly, for

each xi, yi ∈ C0, 1 ≤ i ≤ g − 1 − k there is a chain of smooth, genus zero curves in B′ which

meets C0 at xi, yi. This chain contains precisely one component Ci which is mapped to P10 by

g0. The components C0 and Ci, 1 ≤ i ≤ g − 1− k are the only components mapped to P10.

Suppose for a contradiction [g0, g0] ∈ BN. By definition, we have a 1-dimensional, irreduciblesmooth variety ∆ with chosen point 0 ∈ ∆, and a family of pairs of admissible covers

(g1,t : Bt → Tt, g2,t : Bt → Tt), t ∈ ∆,

such that (g1,0, g2,0) = (g0, g0), and for t 6= 0, Tt ' P1, Bt smooth and g1,t 6= g2,t. The Tt comeswith the data of unlabelled marked points at the branch points. After replacing ∆ by an etalecover, we may label these marked points, so that, on the central fibre, the first three marked


points p1, p2, p3 all lie on the component P10 ⊆ T0. Now forget about all other marked points,

treat (Tt) as a family of semistable genus 0, 3-marked curves, and perform stabilisation on thisfamily. This blows down −1 curves on the central fibre to produce a new family of pairs ofmorphisms

(g′1,t : B′t → P1, g′2,t : B′t → P1), t ∈ ∆,

such that the general fibre is unchanged, but the special fibre is now a morphism with smoothtarget. By stabilising unstable components on the domain on the central fibre, we arrive at afamily of stable maps

(h1,t : B′′t → P1, h2,t : B′′t → P1), t ∈ ∆,

such that hi,t = gi,t for t 6= 0, i = 1, 2.The stable map h := h1,0 = h2,0 is easy to describe from the properties of the admissible cover

g0 : B′ → T ′ given above. The curve B′′0 consists of the smooth, hyperelliptic curve C0 together

with rational components Ri meeting C0 precisely at xi, yi for 1 ≤ i ≤ g− 1− k, as in Figure 2.Let B′′ → ∆ be the total family with fibre over t given by B′′t . Then B′′ has isolated sin-

gularities over the nodes of the central fibre B′′0 . We have two line bundles, L1 and L2, on B′′with Li,t ' h∗i,t(OP1(1)) for i = 1, 2. Consider N := L1 ⊗ L2. By assumption, for t 6= 0 the two

morphisms h1,t : Bt = B′′t → P1 and h2,t : Bt = B′′t → P1 are distinct, so h0(B′′t ,Nt) ≥ 4.Assume firstly B′′ is smooth. The rational components Ri of the central fibre define Cartier

divisors on B′′ for 1 ≤ i ≤ g − 1− k. Consider the line bundle

N ′ := N (

g−1−k∑i=1

Ri).

For t 6= 0, N ′t ' Nt. On the other hand, the line bundle N ′0 on the central fibre B′′0 satisfies thehypothesis of Proposition 3.11, so that h0(B′′0 ,N ′0) = 3. This contradicts semicontinuity.

In the general case, blow B′′ up over the nodes on the central fibre to obtain a smooth surface

B → ∆. This now introduces chains

Γxi = Γxi,1 ∪ . . . ∪ Γxi,li , Γyi = Γyi,1 ∪ . . . ∪ Γyi,mi

of rational curves into the central fibre for all 1 ≤ i ≤ g − 1− k, where Γxi,j resp. Γyi,1 are the

components of Γxi resp. Γyi , and Γxi resp. Γyi meets the component C0 precisely at xi resp.yi. If the two chains have different lengths, say li < mi, then we increase the length of Γxias follows. First, blow up B′′ at xi. The central fibre is no longer reduced, so follow stablereduction by performing a degree two base change and then normalizing, [ACGHII] pg. 106ff.This has the effect of increasing the length of the chain Γxi by one (the total family remainssmooth). By repeating this procedure, we may assume li = mi. Order the components such that

Γxi,1 resp. Γyi,1 are the unique components meeting C0 and such that Γxi,j resp. Γyi,j intersectsΓxi,j+1 and Γxi,j−1 resp. Γyi,j+1 and Γyi,j−1 at one point, and intersects no other components,for 2 ≤ j ≤ c− 1.

Let N ′ denote the pull-back of N to B′′. Let

Zi := (Γxi,1 + Γyi,1) + 2(Γxi,2 + Γyi,2) + . . .+ c(Γxi,c + Γyi,c) + (c+ 1)Ci

for 1 ≤ i ≤ g − 1− k. Define N ′′ := N ′(g−1−k∑i=1

Zi). Then one checks that

N ′′|Γxi' OΓxi

, N ′′|Γyi' OΓyi

, N ′′|Ci ' OΓCi

whereas N ′′|C0' A2 +

g−1−k∑i=1

(xi + yi). We now reach a contradiction from Remark 3.12.


4. Extending the Syzygy Divisor

In this section we construct an extension of the Koszul divisor. We keep the notation of theprevious section and set n = g+ 1− 2k. We further assume k ≥ 3 (and thus g ≥ 5) throughout.

In order to construct the extended divisor, it is convenient to work with stable maps. Let

H†,nsg,d denote the moduli space of finite stable maps f : C → P1 of degree d such that C has

genus g and has no disconnecting nodes and further with h0(C, f∗OP1(1)) = 2. We let

Hnsg,d := H†,nsg,d /PGL(2)

be the space obtained by modding out by the PGL(2) action on the target. LetM†g,d(P1) be the

moduli space of stable maps f : C → P1 with f∗[C] = d[P1]. Then Hnsg,d is an open subset of the

compact space Mg,d(P1) := M†g,d(P1)/PGL(2) of stable maps f : C → P1 with f∗[C] = d[P1],

modulo isomorphisms of the target.

If Hg,d ⊆ Hnsg,d denotes the open locus of finite stable maps f : C → P1 with C irreducible

and h0(C, f∗OP1(1)) = 2, then the complement of Hg,d has codimension two in Hnsg,d, see [DP].

We furthermore let Hg,d denote the open locus of stable maps f : C → P1 with C smooth and

h0(C, f∗OP1(1)) = 2. The boundary of Hnsg,d is divisorial, with three components T , D, ∆, [DP,

§2B]. The generic point of T corresponds to a stable map f : C → P1 with C smooth such thatf has a branch point q with

f−1(p) = 3p1 + p2 + . . .+ pd−2

for distinct points p1, . . . , pd−2. The divisor D is defined similarly, with the exception that thegeneric stable map now has a branch point q with

f−1(p) = 2p1 + 2p2 + . . .+ pd−2.

The divisor ∆ parametrizes all stable maps f : C → P1 with C singular.

We will construct the extended Koszul divisor

Kos ⊆ Hnsg+n,k+n

by studying the minimal free resolutions of scrolls attached to a finite morphism [f : C → P1] ∈Hnsg,d. As f is finite and flat, f∗OC is a vector bundle of rank k+n on P1. We have f∗O∨C = f∗ωf ,

where ωf ' ωC ⊗ (f∗ωP1)∗ is the relative canonical bundle. Dualising the natural morphismOP1 → f∗OC leads to an exact sequence

0→ Ef → f∗ωf → OP1 → 0,

for a rank k + n− 1 vector bundle Ef , called the Tschirnhausen bundle of f .

Let α : P(Ef ) → P1 be the associated projective bundle. Then the morphism f∗(Ef ) → ωfis surjective, and furthermore it induces a natural, closed immersion i : C ⊆ P(Ef ) such thatα i = f , [CE]. Tensoring the morphism f∗(Ef ) → ωf by f∗OP1(−2) = f∗ωP1 produces amorphism f∗(Ef (−2)) ωC . We thus get a morphism

j : C → P(Ef (−2)),

which is, in fact, a closed immersion.From the short exact sequence 0→ OP1 → f∗OC → E∨f → 0, together with h0(C, f∗OP1(1)) =

2, we deduce h0(P1, (Ef (−2))∨(−1)) = 0. As Ef (−2) splits as a sum of line bundles, we see Ef (−2)is globally generated. Further, deg det(Ef (−2)) = g − k + 1. We thus have a morphism

q : P(Ef (−2))→ P(H0(P1, Ef (−2))) ' Pg+n−1


such that the composite j q : C → Pg+n−1 is the canonical morphism, cf. [Sch1]. In particular,ωC is globally generated. Also observe that if h0(C, f∗(OP1(2))) = 3, each line bundle in thedecomposition of Ef (−2)) is very ample and then q is a closed immersion.

Let α′ : P(Ef (−2)) → P1 be the projection. The Picard group of P(Ef (−2)) is generated by

the class of R := α′∗(OP1(1)) together with H := q∗(OPg+n−1(1)). Using the Eagon–Northcott

complex, one can explicitly describe the minimal free resolution of

ΓH :=⊕q∈Z

H0(P(Ef (−2)), qH),

as a SymH0(P(Ef (−2)), H) module, [Sch1]. This gives:

dimKp,0(P(Ef (−2)), H) = 0 for p > 0, whereas dimK0,0(P(Ef (−2)), H) = 1,

dimKp,1(P(Ef (−2)), H) = p

(g − k + 1

p+ 1

)for p ≥ 0,

dimKp,q(P(Ef (−2)), H) = 0 for all q ≥ 2 and any p.

(1)

In particular, Kp,1(P(Ef (−2)), H) = 0 for p > g − k, whereas Kg−k,1(P(Ef (−2)), H) = g − k.We record the following lemma for future use.

Lemma 4.1. We have the vanishing H i(OP(Ef (−2))(qH)) = 0 for i ≥ 1, q ≥ 0. Furthermore,

H i(OP(Ef (−2)(−H)) = 0 for i ≥ 2.

Proof. Let Xf be the rational normal scroll given as the image of q : P(Ef (−2))→ Pg+n−1. Thelemma now follows from the fact that Xf has rational singularities and is 2-regular.

Define the kernel bundles MH and MωC by the short exact sequences

0→MH → H0(P(Ef (−2)), H)⊗OP(Ef (−2)) → H → 0

0→MωC → H0(C,ωC)⊗OC → ωC → 0.

As C is linearly normal in P(Ef (−2)), j∗MH 'MωC . For any p ≥ 1, note that

H0(P(Ef (−2)),∧pMH) = H0(C,∧pMωC ) = 0,

[AN, Remark 2.5]. Further, observe the short exact sequences

0→ ∧p+1MH ⊗Hq−1 → ∧p+1H0(P(Ef (−2)), H)⊗Hq−1 → ∧pMH ⊗Hq → 0,(2)

0→ ∧p+1MωC ⊗ ωq−1C → ∧p+1H0(C,ωC)⊗ ωq−1

C → ∧pMωC ⊗ ωqC → 0,(3)

valid for any p ≥ 0, q ∈ Z. We will make use of the following vanishing statement.

Lemma 4.2. We have H i(∧pMH ⊗ qH) = 0 for i ≥ 2, q ≥ 0 and arbitrary p ∈ Z.

Proof. By the short exact sequence (2) and Lemma 4.1, it suffices to show H i−1(∧p−1MH ⊗ (q+1)H) = 0. Continuing in this fashion, it suffices to show H1(∧p−i+1MH ⊗ (q+ i− 1)H) = 0. AsH1((q+i−1)H) = 0, this is equivalent to Kp,q+i(P(Ef (−2)), H) = 0, which holds as q+i ≥ 2.

Recall the description of Koszul cohomology via kernel bundles, [AN, Ch. 2]. Let X be aprojective variety over C (not necessarily irreducible), F a coherent sheaf on X, and L a basepoint free line bundle. Define the kernel bundle ML on X as above by

0→ML → H0(X,L)⊗ L→ L→ 0.

Consider the Koszul group Kp,q(X;F,L). We then have

Kp,q(X;F,L) ' Coker(∧p+1H0(L)⊗H0(F ⊗ Lq−1)→ H0(∧pML ⊗ F ⊗ Lq))' Ker(H1(∧p+1ML ⊗ F ⊗ Lq−1)→ ∧p+1H0(L)⊗H1(F ⊗ Lq−1))

As a first consequence, we have the following well-known lemma.


Lemma 4.3. There is a natural injective morphism

αf : Kg−k,1(P(Ef (−2)), H) → Kg−k,1(C,ωC).

Proof. We have Kg−k,1(P(Ef (−2)), H) ' H1(∧g−k+1MH) as H1(OP(Ef (−2))) = 0, and further

Kg−k,1(C,ωC) ' Ker(H1(∧g−k+1MωC )→ ∧g−k+1H0(ωC)⊗H1(OC)).

We now show the restriction H1(∧g−k+1MH) → H1(∧g−k+1MωC ) is injective. For this, itsuffices to show H1(∧g−k+1MH ⊗ IC) = 0. Now H1(IC) = 0, as C is linearly normal andH1(OP(Ef (−2))) = 0, so this is in turn equivalent to Kg−k,1(P(Ef (−2)); IC , H) = 0. This latter

claim is obvious, as H0(IC(H)) = 0 by linear normality of C.It remains to show the composition H1(∧g−k+1MH)→ H1(∧g−k+1MωC )→ ∧g−k+1H0(ωC)⊗

H1(OC) is zero. But this map factors through the morphism H1(∧g−k+1MH)→ ∧g−k+1H0(H)⊗H1(OP(Ef (−2))) = 0.

The divisor Kos will parametrize those [f : C → P1] ∈ Hnsg,d such that αf is surjective.

Lemma 4.4. The map αf is surjective if and only if the restriction map

βf : H0(∧g−k−1MH ⊗H2)→ H0(∧g−k−1MωC ⊗ ω2C)

is injective.

Proof. The map αf fits into a commutative diagram

0 // ∧g−k+1H0(H)⊗H0(H) //

∼=

H0(∧g−kMH ⊗H) //

resC

Kg−k,1(P(Ef (−2)), H) //

αf

0

0 // ∧g−k+1H0(ωC)⊗H0(ωC) // H0(∧g−kMωC ⊗ ωC) // Kg−k,1(C,ωC) // 0

,

with exact rows. By the snake lemma, surjectivity of αf is equivalent to surjectivity of resC .We have seen that H1(OP(Ef (−2))(H)) = 0 and Kp,2(P(Ef (−2)), H) = 0 for all p. From the

kernel bundle description of Koszul cohomology, H1(∧g−kMH ⊗H) = 0. We have a diagram

0 // H0(∧g−kMH ⊗H) //

resC

∧g−kH0(H)⊗H0(H) //

∼=

H0(∧g−k−1MH ⊗H2) //

βf

0

0 // H0(∧g−kMωC ⊗ ωC) // ∧g−kH0(ωC)⊗H0(ωC) // H0(∧g−k−1MωC ⊗ ω2C)

,

with exact rows. By the snake lemma, surjectivity of resC is equivalent to injectivity of βf .

As Kg−k−1,2(P(Ef (−2)), H) = 0,

im(βf ) ⊆ Ker(H0(∧g−k−1MωC ⊗ ω2C)→ Kg−k−1,2(C,ωC)).

By Koszul duality, Kg−k−1,2(C,ωC) ' Kg−k,1(C,ωC)∨. Composing with α∨f gives a surjection

ψf : H0(∧g−k−1MωC ⊗ ω2C)→ Kg−k,1(P(Ef (−2)), H)∨,

with im(βf ) ⊆ Ker(ψf ).

Lemma 4.5. We have a natural isomorphism Ker(ψf ) ' H2(∧g+1−kMH ⊗ IC)∨.

Proof. Firstly, Kg−k,1(P(Ef (−2)), H)∨ ' H1(∧g−k+1MH)∨ asH1(OP(Ef (−2))) = 0. Serre–Dualitygives isomorphisms

H0(C,∧g−k−1MωC ⊗ ω2C)∨ ' H1(C,∧g−k+1MωC ),


since ∧pMωC ⊗ ωC ' ∧g+n−1−pM∨ωC. We identify ψ∨f with the natural isomorphism

H1(∧g−k+1MH)→ H1(C,∧g−k+1MωC ).

Then Ker(ψf ) ' (Coker(ψ∨f ))∨ ' H2(∧g+1−kMH ⊗ IC)∨ using Lemma 4.2.

Putting the above pieces together, we have a natural map

γf : H0(∧g−k−1MH ⊗H2)→ H2(∧g+1−kMH ⊗ IC)∨,

and Kg−k,1(C,ωC) > g − k if and only if γf fails to be injective. We will see that both sides of

this map have the same dimension. This will allow us to construct Kos as a degeneracy locus ofvector bundles of the same rank.

Lemma 4.6. We have:

h0(∧g−k−1MH ⊗H2) = h2(∧g+1−kMH ⊗ IC) = (2g − 2k)

(2g − 2k + 1

g − k

)− (g − k).

Further, h0(∧g−k−1MωC ⊗ ω2C) = (2g − 2k)

(2g − 2k + 1

g − k

).

Proof. We have H1(∧g−kMH ⊗H) = 0 as Kg−k−1,2(P(Ef (−2)), H) = 0 = H1(H). So

h0(∧g−k−1MH ⊗H2) = (g + n)

(g + n

g − k

)− h0(∧g−kMH ⊗H)

by the short exact sequence (2). We further have a short exact sequence

0→ ∧g−k+1H0(H)→ H0(∧g−kMH ⊗H)→ Kg−k,1(P(Ef (−2)), H)→ 0,

since H0(∧g−k+1MH) = 0. Thus h0(∧g−kMH ⊗ H) = g − k +(g+ng−k+1

). We now deduce

h0(∧g−k−1MH ⊗ H2) = (2g − 2k)(

2g−2k+1g−k

)− (g − k) from n = g + 1 − 2k and the identity(

ab

)=(aa−b).

We now turn our attention to h2(∧g+1−kMH ⊗ IC). We first show Hq(∧g+1−kMH ⊗ IC) = 0for q 6= 2. We have already seen this for q = 1 in the course of the proof of Lemma 4.3. Forq = 0, this follows from the short exact sequence

0→ IC → OP(Ef (−2)) → OC → 0,

and the fact that H0(∧pMH) = 0 for any p. For q ≥ 3, this again follows from the abovesequence, since Hq−1(∧pMH⊗OC) = 0 (as dimC = 1) and Hq(∧pMH) = 0 for any p by Lemma4.2. We therefore have

h2(∧g+1−kMH ⊗ IC) = χ(∧g+1−kMH ⊗ IC) = χ(∧g+1−kMH)− χ(∧g+1−kMωC ).

By Lemma 4.2 and since H0(∧g+1−kMH) = 0,

χ(∧g+1−kMH) = −h1(∧g+1−kMH) = −dimKg−k,1(P(Ef (−2)), H) = −(g − k),

as H1(OP(Ef (−2))) = 0. Next, the short exact sequence (3) and induction on p gives

χ(∧pMωC ⊗ ωqC) = ((2q − 1)(2g − 2k)− 2p)

(2g − 2k

p

)for any p, q ≥ 0, using that g(C) = g + n = 2g − 2k. Thus h2(∧g+1−kMH ⊗ IC) = (2g −2k)(

2g−2k+1g−k

)− (g−k), using

(ab

)= a+1−b

a+1

(a+1b

). The last identity follows since h0(∧g−k−1MωC ⊗

ω2C) = h1(∧g+1−kMωC) = −χ(∧g+1−kMωC ), from the proof of Lemma 4.5.


The construction of the Tschirnhausen bundle relativizes. Namely, if C/S is a flat family ofconnected, nodal curves over a base S, PS is a P1 bundle over S and fS : C → PS is a finite andflat morphism over S, there is a relative Tschirnhausen bundle EfS fitting into the sequence

0→ EfS → fS∗ωfS → OPS→ 0.

Let

C u //

ν ##

P

µ

Hnsg+n,k+n

be the universal cover, where P → Hnsg+n,k+n is a P1 bundle. The universal Tschirnhausen bundle

Eu on P restricts to Ef for each point p = [f : C → P1] ∈ Hnsg,d. We further have the projective

bundle pr : P(Eu ⊗ ωµ)→ P and a closed immersion j : C → P(Eu ⊗ ωµ).

Let µ′ = pr µ : P(Eu ⊗ ωµ)→ Hnsg+n,k+n. Let H be the line bundle given by the dual of the

tautological line bundle on P(Eu⊗ωµ). For a point p ∈ Hnsg+n,k+n corresponding to f : C → P1,the base change Hp ' H is a globally generated line bundle with g + n sections. By Grauert’sTheorem, µ′∗H is a vector bundle. Define the line bundle ξ by the formula ξ := detµ′∗H. Thecanonical morphism µ′∗µ′∗H→ H is furthermore surjective, so define the vector bundle MH as

0→MH → µ′∗µ′∗H→ H→ 0.

Then MH restricts to the kernel bundle MH over p. Note that j is defined by the surjectionu∗(Eu ⊗ ωµ) ωu ⊗ u∗ωµ ' ων , and so H|C ' ων . Set

F1 := µ′∗(∧g−k−1(MH ⊗H2))⊗ ξ∗,

this is a vector bundle of rank (2g − 2k)(

2g−2k+1g−k

)− (g − k) by Lemma 4.6. Set

F2 := µ′∗(∧g−k−1(MH ⊗H2 ⊗OC))⊗ ξ∗,

this is a vector bundle of rank (2g − 2k)(

2g−2k+1g−k

). Restriction to C induces a morphism

β : F1 → F2,

with fibre over p equal to βf . Relative duality gives the isomorphism

R1ν∗(∧

g−k+1MH|C) ' (ν∗(∧g−k+1M∨H|C ⊗ ων))∨ ' F∨2 ,

using detMH ' µ′∗ξ ⊗ H∗. Define F3 as F3 := (R1µ′∗

(∧g−k+1MH))∨, this is a vector bun-

dle of rank g − k by the proof of Lemma 4.3. The dual of the natural morphism ψ∨ :R1µ′∗

(∧g−k+1MH)→ R1ν∗(∧

g−k+1MH|C) induced by restriction gives a morphism

ψ : F2 → F3

with fibre over p equal to ψf . For all p ∈ Hnsg+n,k+n, (ψ β)p = ψf βf = 0. Thus ψ β = 0.

By the proof of Lemma 4.3, for any [f : C → P1] we have h1(∧g+1−kMH⊗IC) = 0 and furtherh2(∧g+1−kMH) = 0 from Lemma 4.2. We get a short exact sequence of vector bundles

0→ R1µ′∗

(∧g−k+1MH)→ R1ν∗(∧

g−k+1MH|C)→ R2

µ′∗(∧g−k+1MH ⊗ IC)→ 0,

where F4 := R2µ′∗

(∧g−k+1MH ⊗ IC) is a vector bundle of rank (2g − 2k)(

2g−2k+1g−k

)− (g − k) by

Lemma 4.6. Thus we may canonically identify

Kerψ ' F∨4


and β induces a morphism γ : F1 → F∨4 with fibre over p = [f ] given by γf . As F1 and F4 havethe same rank, we can define the extended Koszul divisor

Kos ⊆ Hnsg+n,k+n

as the degeneracy locus of γ. By the results of the previous chapter, this is a genuine divisor.

Define Kosns as the union of all components of Kos containing an element p = [f : C → P1]with C smooth. The following lemma is a direct consequence of Theorem 3.14.

Lemma 4.7. Let C be a general smooth curve of gonality k < bg+32 c and genus g. Let (xi, yi),

1 ≤ i ≤ n = g + 1− 2k be general pairs of distinct points, and let D be the nodal curve obtainedby glueing xi to yi for all i. Let B be the semistable curve given as the union of C with n smoothrational curves Ri, 1 ≤ i ≤ n = g + 1− 2k and such that each Ri meets the rest of B preciselyat xi, yi. Let

[f : B → P1] ∈ Hnsg+n,k+n

be a morphism with f|C a degree k cover C → P1, and f|Rian isomorphism. Then [f ] /∈ Kosns.

Proof. Consider the closure Kosns ⊆Mg+n,k+n(P1) in the moduli space of stable maps. We have

the projection π′ :Mg+n,k+n(P1)→Mg+n, as well as the projection π : Admg+n,k+n →Mg+n

from Hurwitz space. Then we have the following equality of closed sets π′(Kosns) = π(Kos),sinceMg+n,k+n(P1) andAdmg+n,k+n are isomorphic over the open set of morphisms with smooth

base. By Theorem 3.14, the point [D] ∈ Mg+n defined by the stabilization of B does not lie in

π′(Kosns), i.e. [f ] /∈ Kosns.

To complete the proof of Theorem 0.3 we need to show that, in the situation of Lemma 4.7,

the point [f ] does not lie in the extended Koszul divisor Kos. Note that [f ] lies in precisely oneof the three boundary divisors, namely the divisor ∆ parametrising covers with singular base.

Hence we need to show that the divisor Kos does not contain the boundary divisor ∆. We dothis in the final chapter, using K3 surfaces.

5. K3 Surfaces and Goneric Curves

We start by considering a K3 surface Xd with Picard group generated by two classes L andE with self intersections given by (L)2 = 4d − 4, (E)2 = 0 and (L · E) = d, for d ≥ 3. Byperforming Picard–Lefschetz transformations and a reflection if necessary, we may ensure thatthe class L is big and nef.

Lemma 5.1. Consider the K3 surface Xd as above. Then L is base point free and E is the classof a smooth elliptic curve.

Proof. We firstly show that L is base point free. As L is big and nef, it suffices to show there is nosmooth elliptic curve F with (L ·F ) = 1. Assume such an F existed, and write F = xL+yE forx, y ∈ Z. As F is smooth and elliptic, (F )2 = 0 which implies 0 = (xL+yE ·F ) = x+y(E ·F ) =x(1 + dy). If x = 0, (L · F ) = yd 6= 1, since d ≥ 2, so dy = −1, which is again impossible. ThusL is base point free.

We next show that E is the class of a smooth elliptic curve. As (E)2 = 0 and E is primitive (bydefinition) it suffices to show that E is nef. Since (E ·L) > 0, and L is big and nef, E is effective.Suppose E were not nef. Then there would be some smooth, rational curve R with (R ·E) < 0.Write R = xL+yE for x, y ∈ Z. Then (R·E) < 0 implies x < 0. As (R)2 = −2 and R is effective,we must have y > 0. We have −2 = (R)2 = (R ·xL+yE) = x(R ·L) +y(R ·E) = x((R ·L) +yd)which is clearly impossible for d ≥ 3.


We now wish to study the Brill–Noether theory of a smooth curve C ∈ |L|. To do this, wefollow closely [K1, §2], which works in the situation of a higher rank Picard lattice containingthe rank two lattice Pic(Xd).

Lemma 5.2. Let D ∈ Pic(Xd) be effective with (D)2 ≥ 0 and assume in addition L − D iseffective and (L−D)2 > 0. Then D = cE for some integer c.

Proof. This is a slight modification of [K1, Lemma 2.5]. Write D = xL + yE. As L − D iseffective and E nef, (L −D · E) = (1 − x)(L · E) ≥ 0 and so x ≤ 1. From (D · E) ≥ 0 we seex ≥ 0. If x = 1, (L−D)2 = y2(E)2 = 0, so we must have x = 0 as required.

The next lemma gives the Clifford index of any smooth curve in the linear system |L|.

Lemma 5.3. Let C ∈ |L| be a smooth curve. Then C has Clifford index d− 2.

Proof. This is essentially the same as [K1, Lemma 2.6].

We can now describe the Brill–Noether locus W 1d (C).

Lemma 5.4. Let C ∈ |L| be smooth. Then W 1d (C) is reduced and zero dimensional. The

underlying set of W 1d (C) consists of the single point OC(E).

Proof. Arguing exactly as in [K1, Lemmas 2.7, 2.8], we see that W 1d (C) is set-theoretically a

single point.It remains to show that W 1

d (C) is reduced. By Theorem 3.4, we need to show h0(OC(2E)) = 3.From the exact sequence

0→ E → 2E → 2E|E ' OE → 0,

we deduce h1(2E) = 1 and then h0(2E) = 3 by Riemann–Roch. By the short exact sequence

0→ 2E − C → 2E → 2E|C → 0,

it suffices to show h0(2E − C) = h1(2E − C) = 0. As (C − 2E)2 = −4, by Riemann–Rochit suffices to show that neither 2E − C nor C − 2E are effective. As (E · 2E − C) < 0 andE is nef, 2E − C is not effective. Now suppose C − 2E was an effective divisor with integralcomponents R1, . . . , Rl for l ≥ 1. For each i, we have Ri = aiL + biE for integers ai, bi, with∑

i ai = 1,∑

i bi = −2. As (E · Ri) ≥ 0, ai ≥ 0 for all i. Thus, without loss of generality, wemay assume a1 = 1 and ai = 0 for i > 1. As Ri is integral, we must then have bi = 1 for i > 1.Thus R1 = L(1 + l)E which implies (R1)2 = 4d − 4 − 2d(1 + l) ≤ −4, contradicting that R1 isintegral.

We can now prove Theorem 0.3.

Proof of Theorem 0.3. Let [f : B → P1] be as in the statement of Lemma 4.7. We firstly claimthat we have an injective map Kg−k,1(C,ωC) → Kg−k,1(B,ωB). This is a standard argument(cf. [V1, Corollary 1]). By the Mayer–Vietoris sequence, we have a natural injection H0(ωC) →H0(ωB) as well as the composition of injections H0(ω2

C) → H0(ω2C

∑i(xi+ yi)) → H0(ω2

B). Wethen get a commutative diagram

∧g−k+1H0(ωC)δ0 //

∧g−kH0(ωC)⊗H0(ωC) //

∧g−k−1H0(ωC)⊗H0(ω2C)

∧g−k+1H0(ωB)δ′0 // ∧g−kH0(ωB)⊗H0(ωB) // ∧g−k−1H0(ωC)⊗H0(ω2

C)

.

The conclusion now follows from the existence of the maps ∧ : ∧g−kH0(ωC) ⊗ H0(ωC) →∧g−k+1H0(ωC) and ∧′ : ∧g−kH0(ωB) ⊗ H0(ωB) → ∧g−k+1H0(ωB) with ∧ δ0 = ±(g − k)Id,∧′ δ′0 = ±(g − k)Id (which have the natural commutativities with the above maps).


We secondly claim that [f ] does not lie in the extended Koszul divisor Kos. In light of theinjective map above, this will complete the proof. As [f ] lies in exactly one boundary divisor,

namely ∆, all that remains is to show that the divisor Kos does not contain ∆. By Lemma 5.4together with Corollary 3.3, we know that, for any C ∈ |L|, a smooth curve on the K3 surfaceXd for d = k + n, n = g + 1 − 2k, we have Kg−k,1(C,ωC) = g − k. By the Lefschetz Theorem[G], the same holds for any integral nodal curve C0 ∈ |L|. As any integral, nodal curve C0 (withat least one node) defines a point in ∆, it suffices to show that such curves exist for the generalXd.

To do this, it suffices to take g + n ≥ 8, as the conclusion of the Theorem is well–knownfor g ≤ 8 by [Sch1]. Then we see that L − E is very ample for Xd general with the givenPicard lattice, by degenerating to the K3 surface YΩg+n from [K1, Lemma 2.3]. Choose a curveC1 ∈ |L − E| meeting a smooth elliptic curve E0 ∈ |E| transversally, and consider the nodalcurve C1 ∪ E0. Pick any node p1 ∈ C1 ∪ E0. Then, by [T, Thm. 3.8], the moduli space V1(Xd)parametrising deformation of C1∪E0 preserving the assigned node p1 is smooth near (C1∪E0, p1)of dimension g + n− 1. As dim |L− E|+ dim |E| = k + n = g + 1− k < g − 1 for k ≥ 3, thereexist integral, nodal curves C0 ∈ |L| with exactly one node, completing the proof.

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Humboldt-Universitat zu Berlin, Institut fur Mathematik, Unter den Linden 610099 Berlin, GermanyE-mail address: [email protected]

Humboldt-Universitat zu Berlin, Institut fur Mathematik, Unter den Linden 610099 Berlin, GermanyE-mail address: [email protected]

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