Syzygies of 5-Gonal Canonical Curves

Christian Bopp

Received: May 7, 2014 Revised: May 7, 2015

Communicated by Gavril Farkas

Abstract. We show that for 5-gonal curves of odd genus g ≥ 13 and even genus g ≥ 28 the ⌈\frac{g−1}{2}⌉-th syzygy module of the curve is not determined by the syzygies of the scroll swept out by the special pencil of degree 5.

2010 Mathematics Subject Classification: 13D02,14H51

Keywords and Phrases: syzygies, canonical curves, gonality, special divisor

1. Introduction

In this article we study minimal free resolutions of the coordinate ring S_{C} of 5-gonal canonically embedded curves C ⊂ ℙ^{g−1}. The gonality of a curve C is defined as the minimal degree of a nonconstant map C⟶ℙ^{1}. A pencil of degree k on a canonically embedded curve C of genus g, defining a degree k map C → ℙ^{1} sweeps out a rational normal scroll X of dimension d = k − 1 and degree f = g − k + 1. It follows that the linear strand of X is a subcomplex of the linear strand of the curve C. To be more precise the scroll contributes with an Eagon-Northcott complex of length g−k to the linear strand of the curve. This means in particular, that the Betti numbers of X give a lower bound for the Betti numbers of C. The main focus of this article lies on the relation between the Betti numbers of the canonical curve C ⊂ ℙ^{g−1} and the Betti numbers of the scroll X defined by a pencil of minimal degree on C. >From the values of the Hilbert function H_{S_{C}} and the relation with Betti numbers (see [Eis05, Corollary. 9.4 and Corollary. 1.10]) one obtains the following relation for the Betti numbers of a canonical curve C ⊂ ℙ^{g−1}:

β_{i,i+1}(C) = i · \binom{g − 2}{i + 1} − (g − i − 1) · \binom{g − 2}{i − 2} + β_{i−1,i+1}(C).

Since the minimal free resolution of a canonical curve is self-dual, we have

β_{i−1,i+1}(C) = β_{g−i−1,g−i}(C) ≥ β_{g−i−1,g−i}(X) and a direct computation for the case i = ⌈\frac{g−3}{2}⌉ shows that if k > 3 and g ≥ 5, then

β_{i,i+1}(C) ≥ i · \binom{g − 2}{i + 1} − (g − i − 1) · \binom{g − 2}{i − 2} + β_{g−i−1,g−i}(X) > β_{i,i+1}(X)

for all i = 1, …, ⌈\frac{g−3}{2}⌉. We are interested in the Betti numbers β_{i,i+1}(C) for i ≥ ⌈\frac{g−1}{2}⌉. The gonality of a general canonical curve of genus g is precisely ⌈\frac{g+2}{2}⌉ and therefore β_{n,n+1}(X) = 0, where n = ⌈\frac{g−1}{2}⌉. For odd genus g and ground field of characteristic 0, Voisin and Hirschowitz-Ramanan (see [Voi05] and [HR98]) showed that the locus

K_{g} := {C ∈ M_{g} | β_{n,n+1}(C) ≠ 0}

defines an effective divisor in the moduli space of curves, the so-called Koszul divisor. On the Hurwitz-scheme H_{g,k} a natural analogue of the Koszul divisor could be the following

K_{g,k} := {C ∈ H_{g,k} | β_{n,n+1}(C) > β_{n,n+1}(X)}.

If the genus is odd, then K_{g,k} is a divisor on the Hurwitz-scheme if β_{n,n+1}(C) = β_{n,n+1}(X) holds for a general curve C ∈ H_{g,k} (see [HR98, §3]). For gonality k = 3, 4 it is known from [Sch86, §6] that the so-called iterated mapping cone construction, which we recall in Section 2, always gives a minimal free resolution of C ⊂ ℙ^{g−1}. In particular β_{n,n+1}(C) = β_{n,n+1}(X) holds for general 3-gonal and 4-gonal canonical curves. For 5-gonal curves we used Macaulay2 (see [GS]) to verify computationally, that β_{n,n+1}(C) = β_{n,n+1}(X) holds for general 5-gonal canonical curves of genus g < 13. We will show that K_{g,5} is no longer a divisor for odd g ≥ 13 by proving the following theorem.

Theorem. Let C be a 5-gonal canonical curve of genus g and n = ⌈\frac{g−1}{2}⌉. Then

β_{n,n+1}(C) > β_{n,n+1}(X)

for odd genus g ≥ 13 and even genus g ≥ 28.

The proof is based on the techniques introduced in [Sch86]. First we resolve the curve C as an 𝒪_{ℙ(ℰ)}-module, where ℙ(ℰ) is the bundle associated to the rational normal scroll swept out by the g_{5}^{1}. In the next step we resolve the 𝒪_{ℙ(ℰ)}-modules occurring in this resolution by Eagon-Northcott type complexes. An iterated mapping cone construction then gives a non-minimal resolution of C ⊂ ℙ^{g−1}. By determining the ranks of the maps which give rise to non-minimal parts in the iterated mapping cone we can decide whether the curve has extra syzygies. In the last section we discuss the genus 13 case in detail.

Remark 1.1. The proof of the main theorem does not depend on the characteristic of the ground field 𝕜. However for char(𝕜) > 0 it is possible that β_{n,n+1}(C) > β_{n,n+1}(X) for general 5-gonal curves of genus g < 13. This happens, for example, for 5-gonal curves of genus 7 over a field of characteristic 2 [Sch86, §7].

Some of the statements in this article rely on computations done with Macaulay2 [GS]. The Macaulay2 code which verifies these statements can be found here: http://www.math.uni-sb.de/ag-schreyer/images/data/computeralgebra/ fiveGonalFile.m2

2. Scrolls, Pencils and Canonical Curves

In this section we briefly summarize the connections between pencils on canonical curves and rational normal scrolls. Most of this section follows Schreyer’s article [Sch86].

Definition 2.1. Let e_{1} ≥ e_{2} ≥ ⋯ ≥ e_{d} ≥ 0 be integers, ℰ = 𝒪_{ℙ^{1}}(e_{1}) ⊕ ⋯ ⊕ 𝒪_{ℙ^{1}}(e_{d}) and let π : ℙ(ℰ) → ℙ^{1} be the corresponding ℙ^{d−1}-bundle. A rational normal scroll X = S(e_{1}, …, e_{d}) of type (e_{1}, …, e_{d}) is the image of

j : ℙ(ℰ) → ℙH^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(1)) = ℙ^{r}

where r = f + d − 1 with f = e_{1} + ⋯ + e_{d} ≥ 2.

In [Har81, §3] it is shown that the variety X defined above is a non-degenerate d-dimensional variety of minimal degree f = r − d + 1 = codimX + 1. If e_{1}, …, e_{d} > 0, then j : ℙ(ℰ) → X ⊂ ℙH^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(1)) = ℙ^{r} is an isomorphism. Otherwise it is a resolution of singularities and since the singularities of X are rational, we can consider ℙ(ℰ) instead of X for most cohomological considerations. It is furthermore shown that the Picard group Pic(ℙ(ℰ)) is generated by the ruling R = [π^{∗}𝒪_{ℙ^{1}}(1)] and the hyperplane class H = [j^{∗}𝒪_{ℙ^{r}}(1)] with intersection products

H^{d} = f, H^{d−1} · R = 1, R^{2} = 0.

Remark 2.2 ([Sch86, (1.3)]). For a ≥ 0 we have an isomorphism H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(aH + bR)) ≅ H^{0}(ℙ^{1}, S_{a}(ℰ)(b)), where S_{a}(ℰ) denotes the a^{th} symmetric power of ℰ. Thus we can compute the cohomology of the line bundle 𝒪_{ℙ(ℰ)}(aH + bR) explicitly. If we denote by 𝕜[s, t] the coordinate ring of ℙ^{1} and by φ_{i} ∈ H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(H − e_{i}R)) the basic sections, then we can identify sections ψ ∈ H^{0}(𝒪_{ℙ(ℰ)}, 𝒪_{ℙ(ℰ)}(aH + bR)) with homogeneous polynomials

ψ = ∑_{α} P_{α}(s, t)φ_{1}^{α_{1}} … φ_{d}^{α_{d}}

of degree a = α_{1} + ⋯ + α_{d} in φ_{i}’s and with polynomial coefficients P_{α} ∈ 𝕜[s, t] of degree deg P_{α} = α_{1}e_{1} + ⋯ + α_{d}e_{d} + b. Thus for a · min{e_{i}} + b ≥ −1 we get

h^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(aH + bR)) = f\binom{a + d − 1}{d} + (b + 1)\binom{a + d − 1}{d − 1}.

In particular 𝒪_{ℙ(ℰ)}(aH + bR) is globally generated if and only if a ≥ 0 and a · min{e_{i}} + b ≥ 0.

Next we want to describe how to resolve line bundles 𝒪_{ℙ(ℰ)}(aH + bR) by 𝒪_{ℙ^{r}}- modules. If we denote by Φ the 2×f matrix with entries in H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(H)) obtained from the multiplication map

H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(R)) ⊗ H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(H − R)) ⟶ H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(H)),

then the equations of X are given by the 2 × 2 minors of Φ. We define

F := H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(H − R)) ⊗ 𝒪_{ℙ^{r}} = 𝒪_{ℙ^{r}}^{f}

and

G := H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(R)) ⊗ 𝒪_{ℙ^{r}} = 𝒪_{ℙ^{r}}^{2}

and regard Φ as a map Φ : F (−1) → G. For b ≥ −1, we consider the Eagon-Northcott type complex 𝒞 (see [Eis95, Appendix A2.6]), whose j^{th} term is defined by

𝒞_{j}^{b} = \begin{cases}⋀^{j} F ⊗ S_{b−j}G ⊗ 𝒪_{ℙ^{r}}(−j), & for 0 ≤ j ≤ b \\ ⋀^{j+1} F ⊗ D_{j−b−1}G^{∗} ⊗ 𝒪_{ℙ^{r}}(−j − 1), & for j ≥ b + 1 \end{cases}

where S_{j}G denotes the j^{th} symmetric power and D_{j}G^{∗} denotes the j^{th} divided power of G. The differentials δ_{j} : 𝒞_{j}^{b} → 𝒞_{j−1}^{b} are given by the multiplication with Φ for j ≠ b + 1 and by ⋀^{2} Φ for j = b + 1.

Theorem 2.3. The Eagon-Northcott type complex 𝒞^{b}(a) := 𝒞^{b} ⊗ 𝒪_{ℙ^{r}}(a), defined above, gives a minimal free resolution of 𝒪_{ℙ(ℰ)}(aH + bR) as an 𝒪_{ℙ^{r}}- module.

Proof. See [Sch86, §1]. □

Now let C ⊂ ℙ^{g−1} be a canonically embedded curve of genus g and let further

g_{k}^{1} = {D_{λ}}_{λ∈ℙ^{1}} ⊂ |D|

be a pencil of divisors of degree k. If we denote by \overline{D_{λ}} ⊂ ℙ^{g−1} the linear span of the divisor, then

X = ⋃_{λ∈ℙ^{1}} \overline{D_{λ}} ⊂ ℙ^{g−1}

is a (k − 1)-dimensional rational normal scroll of degree f = g − k + 1 (see e.g. [EH87, Theorem 2]). Conversely if X is a rational normal scroll of degree f containing a canonical curve, then the ruling on X cuts out a pencil of divisors {D_{λ}} ⊂ |D| such that h^{0}(C, ω_{C} ⊗ 𝒪_{C}(D)^{−1}) = f.

Notation 2.4. During the rest of this article C ⊂ ℙ^{g−1} will denote a canonical curve with a basepoint free g_{k}^{1}. The variety X = S(e_{1}, …, e_{d}) is the scroll of degree f = g − k + 1 and dimension d = k − 1 defined by this pencil and ℙ(ℰ) will denote the ℙ^{d−1}-bundle corresponding to X.

The next important theorem due to Schreyer explains how to obtain a free resolutions of k-gonal canonical curves C ⊂ ℙ^{g−1} by an iterated mapping cone construction.

Theorem 2.5. i) C ⊂ ℙ(ℰ) has a resolution F_{•} of type

0 ⟶ 𝒪_{ℙ(ℰ)}(−kH + (f − 2)R) ⟶ ∑_{j=1}^{β_{k−3}} 𝒪_{ℙ(ℰ)}(−(k − 2)H + a_{j}^{(k−3)}R) ⟶

… ⟶ ∑_{j=1}^{β_{1}} 𝒪_{ℙ(ℰ)}(−2H + a_{j}^{(1)}R) ⟶ 𝒪_{ℙ(ℰ)} ⟶ 𝒪_{C} ⟶ 0

with β_{i} = \frac{i(k−2−i)}{k−1} \binom{k}{i+1}. ii) The complex F_{•} is self dual, i.e.,

ℋ om(F_{•}, 𝒪_{ℙ(ℰ)}(−kH + (f − 2)R)) ≅ F_{•}

iii) If all a_{k}^{(j)} ≥ −1, then an iterated mapping cone

[[… [𝒞^{(f−2)}(−k) ⟶ ∑_{j=1}^{β_{k−3}} 𝒞^{(a_{j}^{(k−3)})}(−k + 2)] ⟶ …] ⟶ 𝒞^{0}]

gives a, not necessarily minimal, resolution of C as an 𝒪_{ℙ^{g−1}}-module.

Proof. See [Sch86, Corollary 4.4] and [Sch86, Lemma 4.2]. □

Remark 2.6. The a_{i}^{(k)}’s in part i) of the theorem above satisfy certain linear equations obtained from the Euler characteristic of the complex F_{•} and part (ii) of the Theorem above. In particular in [Sch86, §6] it is shown, that for 5-gonal curves C

∑_{i=1}^{5} a_{i} = 2g − 12 and a_{i} + b_{i} = f − 2,

where a_{i} := a_{i}^{(1)}, b_{i} := a_{i}^{(2)} and f = g − 4 is the degree of the rational normal scroll swept out by the g_{5}^{1} on C.

Definition 2.7. We call a partition (e_{1}, …, e_{d}) balanced if max_{i,j} |e_{i}−e_{j}| ≤ 1. A rational normal scroll X = S(e_{1}, …, e_{d}) of dimension d is said to be of balanced type if (e_{1}, …, e_{d}) is a balanced partition. With the same notation as in the remark above we say that a 5-gonal canonical curve satisfies the balancing conditions if

• the scroll X of dimension 4 swept out by the g_{5}^{1} on C is of balanced type and

• the partition (a_{1}, …, a_{5}) of 2g − 12 is balanced.

Remark 2.8. One can show (see e.g. [GV06, Corollary 3.3]) that a generic k-gonal curve sweeps out a scroll of balanced type. The proof uses Ballico’s Theorem [Bal89, Proposition 2.1.1] and the fact that the type of the scroll is uniquely determined by the g_{k}^{1} (see [Sch86, (2.4)]).

For a section Ψ : 𝒪_{X}(−H + bR) ⟶ 𝒪_{X}(aR) in H^{0}(X, 𝒪_{X}(H − (b − a)R)) the induced comparison maps ψ_{•} : 𝒞_{•}^{b}(−1) ⟶ 𝒞_{•}^{a} between the corresponding Eagon-Northcott type complexes are determined by Ψ up to homotopy. For example

Hom(𝒞_{a+1}^{b}(−1), 𝒞_{a+2}^{a}) = Hom(𝒞_{a}^{b}(−1), 𝒞_{a+1}^{a}) = 0

by degree reasons, and therefore the (a+1)^{st}−comparison map ψ_{a+1} is uniquely determined by Ψ (not only up to homotopy). The following lemma is due to Martens and Schreyer.

Lemma 2.9. If Hom(𝒞_{j}^{b}(−1), 𝒞_{j+1}^{a}) = Hom(𝒞_{j−1}^{b}(−1), 𝒞_{j}^{a}) = 0, then the j^{th}- comparison map ψ_{j} : 𝒞_{j}^{b} → 𝒞_{j}^{a} is given (up to a scalar factor) by the composition

ψ_{j} : ⋀^{j} F ⊗ S_{b−j}G ⟶ ⋀^{j} F ⊗ S_{b−j}G ⊗ S_{j−a−1}G ⊗ D_{j−a−1}G^{∗}

⟶^{id⊗mult⊗id} ⋀^{j} F ⊗ S_{b−a−1}G ⊗ D_{j−a−1}G^{∗} ⟶^{id⊗Ψ⊗id} ⋀^{j} F ⊗ F ⊗ D_{j−a−1}G^{∗}

⟶^{∧⊗id} ⋀^{j+1} F ⊗ D_{j−a−1}G^{∗}.

Proof. In [MS86, Lemma of the Appendix] this is shown for the (a + 1)^{st} comparison map but the proof immediately generalizes to our situation as long as Hom(𝒞_{j}^{b}(−1), 𝒞_{j+1}^{a}) = Hom(𝒞_{j−1}^{b}(−1), 𝒞_{j}^{a}) = 0. □

3. Proof of the Main Theorem

The aim of this section is to prove Theorem 3.1 and then to show that a general 5-gonal curve satisfies the balancing conditions.

Theorem 3.1. Let C ⊂ ℙ^{g−1} be a general 5-gonal canonical curve of genus g satisfying the balancing conditions. If X is the scroll swept out by the g_{5}^{1} and n = ⌈\frac{g−1}{2}⌉, then

β_{n,n+1}(C) > β_{n,n+1}(X) for odd g ≥ 13 and even g ≥ 28.

Throughout this section, C ⊂ ℙ^{g−1} will be a 5-gonal canonical curve of genus g. In this case X is a d = 4 dimensional rational normal scroll of degree f = g −4. Recall from Theorem 2.5 and Remark 2.6 that C ⊂ ℙ(ℰ) has a resolution of the form

𝒪_{ℙ(ℰ)}(−5H + (f − 2)R) → ∑_{i=1}^{5} 𝒪_{ℙ(ℰ)}(−3H + b_{i}R)

⟶^{Ψ} ∑_{i=1}^{5} 𝒪_{ℙ(ℰ)}(−2H + a_{i}R) → 𝒪_{ℙ(ℰ)} → 𝒪_{C}

where ∑_{i=1}^{5} a_{i} = 2g − 12, a_{i} + b_{i} = f − 2. The matrix Ψ is skew symmetric by the structure theorem for Gorenstein ideals in codimension 3 and the 5 Pfaffians of Ψ generate the ideal of C (see [BE77, Theorem 2.1]).

As in Theorem 2.3, we denote by F = H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(H −R))⊗𝒪_{ℙ^{g−1}} ≅ 𝒪_{ℙ^{g−1}}^{f} and by G = H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(R))⊗𝒪_{ℙ^{g−1}} ≅ 𝒪_{ℙ^{g−1}}^{2}. By abuse of notation, we will also refer to the vector spaces H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(H −R)) and H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(R)) by F and G, respectively. Resolving the 𝒪_{ℙ(ℰ)}-modules occurring in the minimal resolution of C ⊂ ℙ(ℰ), we get

\xymatrix{∑_{i=1}^{5} 𝒪_{ℙ(ℰ)} (−3H + b_{i}R) \ar[r]^{Ψ} & ∑_{i=1}^{5} 𝒪_{ℙ(ℰ)} (−2H + b_{i}R) \\ ⋮ \ar[u] &  & ⋮ \ar[ul] \\ ∑_{i=1}^{5} 𝒞_{n−2}^{b_{i}}(−3) \ar[u] \ar[rr]^{ψ_{n−2}} &  & ∑_{i=1}^{5} 𝒞_{n−2}^{a_{i}}(−2) \ar[u] \\ ∑_{i=1}^{5} 𝒞_{n−1}^{b_{i}}(−3) \ar[u] \ar[rr]^{ψ_{n−1}} &  & ∑_{i=1}^{5} 𝒞_{n−1}^{a_{i}}(−2) \ar[u] \\ ⋮ \ar[u] &  & ⋮ \ar[u]}

where rank(∑_{i=1}^{5} 𝒞_{n−2}^{b_{i}}) ≤ rank(∑_{i=1}^{5} 𝒞_{n−2}^{a_{i}}) (with equality for odd genus). Next note that if C is a canonical curve of odd genus g = 2n + 1 satisfying the balancing conditions, then we obtain the following inequality

min\{b_{i}\} = ⌊\frac{∑_{i=1}^{5} b_{i}}{5} ⌋ = ⌊\frac{6n − 15}{5}⌋ ≥ n − 2 ≥ ⌈\frac{4n − 10}{5}⌉ = max\{a_{i}\} (3.1)

if n ≥ 5. For even genus g = 2n and n ≥ 8 we similarly get

min\{b_{i}\} = ⌊\frac{6n − 18}{5}⌋ ≥ n − 2 ≥ ⌈\frac{4n − 12}{5}⌉ = max\{a_{i}\}.

It follows that in these cases

𝒞_{n−2}^{b_{i}}(−3) = ⋀^{n−2} F ⊗ S_{b_{i}−n+2}G(−n − 1)

and

𝒞_{n−2}^{a_{i}}(−2) = ⋀^{n−1} F ⊗ D_{n−a_{i}−3}G^{∗}(−n − 1).

Thus, if C is a canonical curve of odd genus g ≥ 11 or even genus g ≥ 16, then the (n − 2)^{th} comparison map in the diagram above

ψ := ψ_{n−2} : ∑_{i=1}^{5} 𝒞_{n−2}^{b_{i}}(−3) ⟶ ∑_{i=1}^{5} 𝒞_{n−2}^{a_{i}}(−2)

has entries in the field 𝕜 and its rank determines the Betti number β_{n,n+1}(C). To be more precise, we have

β_{n,n+1}(C) = β_{n,n+1}(X) + dim ker(ψ),

where the Betti number β_{n,n+1}(X) is given by rank(𝒞_{n}^{0}) = n · \binom{f}{n+1}.

For the proof of Theorem 3.1 we restrict ourselves to curves of odd genus since the theorem is proved in exactly the same way for even genus. For curves of odd genus g = 2n + 1, we distinguish 5 types of curves satisfying the balancing conditions. These types depend on the congruence class of n modulo 5, i.e., on the block structure of the skew symmetric matrix Ψ. Setting r := ⌊\frac{n}{5}⌋ we have the following five possibilities.

Type I: (a_{1}, …, a_{5}) = (a, a, a, a, a), (b_{1}, …, b_{5}) = (b, b, b, b, b) ⇔ a =

4r + 2, b = 6r + 3 and n = 5r + 5.

Type II: (a_{1}, …, a_{5}) = (a − 1, a, a, a, a), (b_{1}, …, b_{5}) = (b + 1, b, b, b, b) ⇔

a = 4r − 1, b = 6r − 2 and n = 5r + 1.

Type III: (a_{1}, …, a_{5}) = (a − 1, a − 1, a, a, a), (b_{1}, …, b_{5}) = (b + 1, b +

1, b, b, b) ⇔ a = 4r, b = 6r − 1 and n = 5r + 2.

Type IV: (a_{1}, …, a_{5}) = (a, a, a, a + 1, a + 1), (b_{1}, …, b_{5}) = (b, b, b, b −

1, b − 1) ⇔ a = 4r, b = 6r + 1 and n = 5r + 3.

Type V: (a_{1}, …, a_{5}) = (a, a, a, a, a + 1), (b_{1}, …, b_{5}) = (b, b, b, b, b − 1) ⇔

a = 4r + 1, b = 6r + 2 and n = 5r + 4.

Proof of Theorem 3.1. Since the proof of the theorem is similar for all different types above, we will only carry out the proof for curves of type II, leaving the other cases to the reader. We show that the map

ψ : (𝒞_{n−2}^{(b+1)} ⊕ ∑_{i=1}^{4} 𝒞_{n−2}^{b})(−3) ⟶ (𝒞_{n−2}^{(a−1)} ⊕ ∑_{i=1}^{4} 𝒞_{n−2}^{a})(−2)

induced by the skew-symmetric matrix

Ψ : ⊕_{𝒪_{ℙ(ℰ)} (−3H+bR)^{⊕4}}^{𝒪_{ℙ(ℰ)} (−3H+(b+1)R)} ⟶ ⊕_{𝒪_{ℙ(ℰ)} (−2H+aR)^{⊕4}}^{𝒪_{ℙ(ℰ)} (−2H+(a−1)R)}

has a non-trivial decomposable element in the kernel. Note that the map

Ψ_{(11)} : 𝒪_{ℙ(ℰ)}(−3H + (b + 1)R) ⟶ 𝒪_{ℙ(ℰ)}(−3H + (a − 1)R)

is zero by the skew-symmetry of Ψ. Thus it is sufficient to find an element in the kernel of the map ψ_{(41)} : 𝒞_{n−2}^{(b+1)}(−3) ⟶ ∑_{i=1}^{4} 𝒞_{n−2}^{(a)}(−2), induced by the submatrix

Ψ_{(41)} : 𝒪_{ℙ(ℰ)}(−3H + (b + 1)R) ⟶ 𝒪_{ℙ(ℰ)}(−2H + aR)^{⊕4}

of the matrix Ψ. By Lemma 2.9, the map ψ_{(41)} is uniquely determined and is given as the composition

𝒞_{n−2}^{b+1}(−3) = ⋀^{n−2} F ⊗ S_{b−n+3}G = ⋀^{n−2} F ⊗ S_{n−a−2}G

↪ ⋀^{n−2} F ⊗ S_{n−a−2}G ⊗ S_{n−a−3}G ⊗ D_{n−a−3}G^{∗}

⟶ ⋀^{n−2} F ⊗ S_{2n−2a−5}G ⊗ D_{n−a−3}G^{∗} ⟶^{id⊗Ψ_{(41)}⊗id} ⋀^{n−2} F ⊗ F^{⊕4} ⊗ D_{n−a−3}G^{∗}

⟶^{∧⊗id} ( ⋀^{n−1} F )^{⊕4} ⊗ D_{n−a−3}G^{∗} = ∑_{i=1}^{4} 𝒞_{n−2}^{a}(−2) .

Since the multiplication map S_{n−a−2}G ⊗ S_{n−a−3}G ⟶ S_{2n−2a−5}G is not injective, we show that the existence of an f ∈ ⋀^{n−2} F and a g ∈ S_{n−a−2}G such that f ∧ Ψ_{(41)}(g · g′) = 0 for all g′ ∈ S_{n−a−3}G. To this end, let g ∈ S_{n−a−2}G be an arbitrary element and let {g′_{1}, …, g′_{n−a−2}} be a basis of S_{n−a−3}G. For i = 1, …, (n − a − 2), we define

(f_{1}^{(i)}, f_{2}^{(i)}, f_{3}^{(i)}, f_{4}^{(i)})^{t} := Ψ_{(41)}(g · g′_{i}) ∈ F^{4}

and choose a maximal linearly independent subset {f_{k}}_{k=1,...,p} of {f_{j}^{(i)}} ⊂ F. Since

n − 2 = 5r − 1 ≥ #{f_{j}^{(i)}} = 4 · dim_{𝕜}(S_{n−a−3}G) = 4(n − a − 2) = 4r
≥ #{f_{k}} = p

holds for all r ≥ 1 (i.e. g ≥ 13), we find a nonzero element f of the form

f = f_{1} ∧ f_{2} ∧ ⋯ ∧ f_{p} ∧ f̃ ∈ ⋀^{n−2} F

for some f̃ ∈ ⋀^{n−p−2} F. By construction, f ⊗ g is in the kernel of ψ_{(41)}, and hence (f ⊗ g, 0, 0, 0, 0)^{t} lies in the kernel of ψ. □

We can immediately generalize Theorem 3.1.

Theorem 3.2. Let C ⊂ ℙ^{g−1} be a general 5-gonal canonical curve satisfying the balancing conditions. Then

β_{n+c,n+c+1}(C) > β_{n+c,n+c+1}(X) for odd genus g = 2n + 1 ≥ 30c + 13
β_{n+c,n+c+1}(C) > β_{n+c,n+c+1}(X) for even genus g = 2n ≥ 30c + 28.

Proof. With the above notation we always have

rank(∑_{i=1}^{5} 𝒞_{n−2+c}^{b_{i}}) ≤ rank(∑_{i=1}^{5} 𝒞_{n−2+c}^{a_{i}}).

Since the curve C satisfies the balancing conditions, we obtain as in (3.1)

min{b_{i}} ≥ n − 2 + c ≥ max{a_{i}}

for odd genus g = 2n+1 if n ≥ 5c+5 or even genus g = 2n if n ≥ 5c+8. Thus, it follows that the Betti number β_{n+c,n+c+1}(C) is determined by the rank of the map

ψ_{n−2+c} : ∑_{i=1}^{5} 𝒞_{n−2+c}^{b_{i}}(−3) ⟶ ∑_{i=1}^{5} 𝒞_{n−2+c}^{a_{i}}(−2)

which has entries in the ground field 𝕜. By the same construction as in the proof of Theorem 3.1, we find a decomposable element in the kernel of ψ_{n−2+c} if r = ⌊\frac{n}{5}⌋ ≥ 3c + 1 (for odd genus) or r = ⌊\frac{n}{5}⌋ ≥ 3c + 3 (for even genus). This gives precisely the range for the genus, as stated in the theorem. □

Next we want to show that the generic 5-gonal curve satisfies the balancing conditions. The balancing conditions on a 5-gonal canonical curve are open conditions. It is therefore sufficient to find an example for each g.

Proposition 3.3. For any odd g ≥ 13 (and even g ≥ 28), there exists a smooth and irreducible 5-gonal canonical genus g curve satisfying the balancing conditions.

Proof. We illustrate the proof for odd genus curves of type II: To this end, let X = S(e_{1}, …, e_{4}) ≅ ℙ(ℰ) with e_{1} ≥ ⋯ ≥ e_{4} be a fixed 4dimensional rational normal scroll of balanced type and of degree f = g − 4. Let further

(a_{1}, …, a_{5}) = (a − 1, a, a, a, a) and (b_{1}, …, b_{5}) = (b + 1, b, b, b, b)

be balanced partitions such that ∑_{i=1}^{5} a_{i} = 2g − 12, g = 2n + 1, n = 5r + 1, a = 4r−1 and a_{i}+b_{i} = g−6. We consider a general skew-symmetric morphism

\underbrace{⊕_{𝒪_{ℙ(ℰ)} (−3H+bR)^{⊕4}}^{𝒪_{ℙ(ℰ)} (−3H+(b+1)R)}}_{=:ℱ} ⟶^{Ψ} \underbrace{⊕_{𝒪_{ℙ(ℰ)} (−2H+aR)^{⊕4}}^{𝒪_{ℙ(ℰ)} (−2H+(a−1)R)}}_{=:ℱ^{∗}⊗ℒ}

If ⋀^{2} ℱ^{∗} ⊗ ℒ is globally generated, then it follows by a Bertini type theorem (see e.g. [Oko94, §3]) that the scheme Pf(Ψ) cut out by the Pfaffians of Ψ is smooth of codimension 3 or empty. Recall from Remark 2.2, that a line bundle 𝒪_{ℙ(ℰ)}(H + cR) is globally generated if and only if e_{4} + c ≥ 0. Thus since for r ≥ 1

min\{e_{i}\} = ⌊\frac{g − 4}{4}⌋ = ⌊\frac{10r − 1}{4}⌋ = 2r + ⌊\frac{2r − 1}{4}⌋ ≥ (b − a + 1) = 2r,

we conclude that in this case

⋀^{2} ℱ^{∗} ⊗ ℒ = 𝒪_{ℙ(ℰ)}(H − (b − a + 1)R)^{⊕4} ⊕ 𝒪_{ℙ(ℰ)}(H − (b − a)R)^{⊕6}

is globally generated. It follows that C = Pf(Ψ) is smooth of codimension 3 or empty. The iterated mapping cone construction gives a free resolution of C ⊂ ℙ^{g−1} of length g − 2 which is not null-homotopic. Therefore it follows from the Auslander-Buchsbaum formula that C is a non-empty (and therefore smooth) arithmetically Cohen-Macaulay scheme. In particular we have a surjective map

\underbrace{H^{0}(ℙ^{g−1}, 𝒪_{ℙ^{g−1}})}_{≅𝕜} → H^{0}(C, 𝒪_{C}) → 0

and therefore C is smooth and connected and hence an irreducible curve. Doing the same for curves of type I, III, IV and V and the even genus cases the result follows for all genera except for g = 15 (in this case ⋀^{2} ℱ^{∗} ⊗ ℒ is not globally generated). For the g = 15 case one can verify the statement by using Macaulay2 (see [GS]). □

Corollary 3.4. K_{g,5} = {C ∈ H_{g,5} | β_{n,n+1}(C) > β_{n,n+1}(X)} equals H_{g,5} for odd genus g ≥ 13 and even genus g ≥ 28.

Proof. For odd genus g ≥ 13 and even genus g ≥ 28 it follows by Theorem 3.1 and Proposition 3.3 that K_{g,5} is a non empty and dense subset of H_{g,5}. The conclusion follows by semi-continuity on the Betti numbers. □

4. A First Example

In this section, we discuss the case of a general 5-gonal canonical curve of genus 13. The rational normal scroll X swept out by the g_{5}^{1} on C is therefore a 4dimensional scroll of type S(3, 2, 2, 2) and degree f = 9. The curve C ⊂ ℙ(ℰ) has a resolution of the form

𝒪_{ℙ(ℰ)}(−5H + 7R) → ⊕_{𝒪_{ℙ(ℰ)} (−3H+4R)^{⊕4}}^{𝒪_{ℙ(ℰ)} (−3H+5R)} ⟶^{Ψ} ⊕_{𝒪_{ℙ(ℰ)} (−2H+3R)^{⊕4}}^{𝒪_{ℙ(ℰ)} (−2H+2R)} → 𝒪_{ℙ(ℰ)} → 𝒪_{C}

where Ψ is a skew-symmetric matrix with entries as indicated below

(4.1)  (Ψ) ∼ ( \begin{matrix}0 & (H − 2R) & (H − 2R) & (H − 2R) & (H − 2R) \\ & 0 & (H − R) & (H − R) & (H − R) \\ & & 0 & (H − R) & (H − R) \\ & & & 0 & (H − R) \\ & & & & 0 \end{matrix} )

We resolve the 𝒪_{ℙ(ℰ)}-modules in the resolution above by Eagon-Northcott type complexes and determine the rank of the maps which give rise to nonminimal parts in the iterated mapping cone. As in Section 3, we denote by F = H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(H − R)) and by G = H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(R)).

[figure: ∑_{i=1}^{5} 𝒞_{3}^{b_{i}}(−3) = ⋀^{3} F ⊗S_{2}G(−6) ; ⊕ ; (⋀^{3} F ⊗S_{1}G(−6))^{⊕4}; ψ_{3}; ∑_{i=1}^{5} 𝒞_{3}^{a_{i}}(−2) = ⋀^{4} F (−6) ; ⊕ ; (⋀^{4} F (−5))^{⊕4}; ∑_{i=1}^{5} 𝒞_{4}^{b_{i}}(−3) = ⋀^{4} F ⊗S_{1}G(−7) ; ⊕ ; (⋀^{4} F (−7))^{⊕4}; ψ_{4}; ∑_{i=1}^{5} 𝒞_{4}^{a_{i}}(−2) = ⋀^{5} F ⊗D_{1}G^{∗}(−7) ; ⊕ ; (⋀^{5} F (−7))^{⊕4}; ∑_{i=1}^{5} 𝒞_{5}^{b_{i}}(−3) = ⋀^{5} F (−8) ; ⊕ ; (⋀^{5} F (−9))^{⊕4}; ψ_{5}; ∑_{i=1}^{5} 𝒞_{5}^{a_{i}}(−2) = ⋀^{6} F ⊗D_{2}G^{∗}(−8) ; ⊕ ; (⋀^{6} F ⊗D_{1}G^{∗}(−8))^{⊕4}]

Note that by degree reasons, the maps indicated above are the only ones which give rise to possibly non-minimal parts in the iterated mapping cone. By the Gorenstein property of canonical curves, it follows that the maps

ψ′_{3} : (⋀^{3} F ⊗ S_{1}G)^{⊕4}(−6) → ⋀^{4} F (−6)

and

ψ′_{5} : ⋀^{5} F (−8) → (⋀^{6} F ⊗ D_{1}G^{∗})^{⊕4}(−8)

are dual to each other and one can easily check the surjectivity of ψ′_{3}. It remains to compute the rank of ψ_{4}.

Proposition 4.1. Let Ψ be a general skew symmetric matrix with entries as indicated above. Then the induced matrix ψ_{4} : ∑_{i=1}^{5} 𝒞_{4}^{b_{i}}(−3) → ∑_{i=1}^{5} 𝒞_{4}^{a_{i}}(−2) has a six dimensional kernel.

Proof. According to Section 2, we can write down the relevant cohomology groups. Let {s, t} be a basis of G = H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(R)) and {φ_{1}} be a basis of H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(H − 3R)) then a basis of H^{0}(ℙ(ℰ), 𝒪_{ℙ(ℰ)}(H − 2R)) is given by {sφ_{1}, tφ_{1}, φ_{2}, φ_{3}, φ_{4}}. We consider the submatrix ψ_{(41)} : ⋀^{4} F ⊗ S_{1}G(−7) →

(⋀^{5} F (−7))^{4} of ψ_{4} induced by the first column of Ψ. As in the proof of Theorem 3.1, the map ψ_{(41)} is given as the composition

\xymatrix{⋀^{4} F ⊗ S_{1}G ≅ ⋀^{4} F ⊗ H^{0}(𝒪_{ℙ(ℰ)}(R)) \ar[d]^{id⊗Ψ_{(41)}} \\ ⋀^{4} F ⊗ H^{0}(𝒪_{ℙ(ℰ)}(H − R))^{⊕4} ≅ ⋀^{4} F ⊗ F^{⊕4} \ar[r]^{∧} & (⋀^{5} F)^{⊕4}.}

By our generality assumption on C, we can assume that the 4 entries of Ψ_{(41)} are independent and after acting with an element in Aut(X), we can furthermore assume that Ψ_{(41)} = (sφ_{1}, φ_{2}, φ_{3}, φ_{4})^{t}. It follows that elements of the form

(λs + μt)sφ_{1} ∧ (λs + μt)φ_{2} ∧ (λs + μt)φ_{3} ∧ (λs + μt)φ_{4} ⊗ (λs + μt), with λ, μ ∈ 𝕜

lie in the kernel of ψ_{(41)}. Expanding those elements we get

λ^{5}s^{2}φ_{1} ∧ sφ_{2} ∧ sφ_{3} ∧ sφ_{4} ⊗ s + ⋯ + μ^{5}stφ_{1} ∧ tφ_{2} ∧ tφ_{3} ∧ tφ_{4} ⊗ t

and conclude that a rational normal curve of degree 5 lies in ℙ(Syz) where Syz ⊂ Tor_{6}^{T} (T/I_{C})_{7} is the subspace of the 6^{th} syzygy module spanned by the extra syzygies and I_{C} ⊂ T denotes the ideal of the canonical curve C. We get β_{6,7}(C) ≥ β_{6,7}(X) + 6 = 222 and by computing one example using Macaulay2, it follows that ψ_{4} has a 6-dimensional kernel in general. □

Remark 4.2. A direct computation using Macaulay2 shows that none of the entries of the skew symmetric matrix Ψ can be made zero by suitable row and column operations respecting the skew symmetric structure of Ψ. By [Sch86, §5] this implies that the 6 extra syzygies are not induced by an additional linear series on C.

The question arises of how the extra syzygies of a 5-gonal canonical curve C ⊂ ℙ^{12} differ from the syzygies induced by the scroll swept out by the g_{5}^{1} on C. At least in the genus g = 13 case we can give an answer in this direction by considering syzygy schemes, originally introduced in [Ehb94].

Definition and Remark 4.3. Let C ⊂ ℙ^{g−1} be a smooth and irreducible canonical curve and let I_{C} ⊂ S be the ideal of C. Let further

L_{•} : S ← S(−2)^{β_{1,2}} ← S(−3)^{β_{2,3}} ← ⋯

be the linear strand of a minimal free resolution of S_{C} = S/I_{C}. For a p^{th} linear syzygy s ∈ L_{p}, let V_{s} be the smallest vector space such that the following diagram commutes

L_{p−1}  ⟵  L_{p}
∪  ∪
V_{s} ⊗ S(−p) ⟵ S(−p − 1) ≅ ⟨s⟩ .

The rank of the syzygy s is defined to be rank(s) := dim V_{s}. Since Hom(L_{•}, S) is a free complex and the Koszul complex is exact, it follows that the maps of the dual diagram extend to a morphism of complexes. Dualizing again we get

\xymatrix{S & L_{1} \ar[l] & ⋯ \ar[l] \ar[l] & L_{p−1} \ar[l] & L_{p} \ar[l] \\ ⋀^{p} V_{s} ⊗ S(−1) \ar[u]_{φ_{p}} & ⋀^{p−1} V_{s} ⊗ S(−2) \ar[l] \ar[u] & ⋯ \ar[l] & V_{s} ⊗ S(−p) \ar[l] \ar[u] & S(−p − 1) . \ar[l] \ar[u]}

By degree reasons there are only trivial homotopies and therefore all the vertical maps except φ_{p} are unique. The syzygy scheme Syz(s) of s is the scheme defined by the ideal

I_{s} = im(S ⟵ ⋀^{p−1} V_{s} ⊗ S(−2)).

The p^{th}-syzygy scheme Syz_{p}(C) of a curve C is defined to be the intersection ⋂_{s∈L_{p}} Syz(s).

Any p^{th}-syzygy of a canonical curve has rank ≥ p + 1 and the syzygies of rank p + 1 are called scrollar syzygies. The name is justified by a theorem due to von Bothmer:

Theorem 4.4 ([GvB07, Corollary 5.2]). Let s ∈ L_{p} be a p^{th} scrollar syzygy. Then Syz(s) is a rational normal scroll of degree p + 1 and codimension p that contains the curve C.

We can now come back to our example of a 5-gonal genus 13 curve. Recall that in this case the space of extra syzygies can be identified with the kernel of the map

𝒞_{4}^{5}(−3) → (𝒞_{4}^{3})^{⊕4}(−2)

which is induced by the first column of the skew symmetric matrix Ψ. We denote by Pf_{1}, …, Pf_{4} ∈ H^{0}(𝒪_{ℙ(ℰ)}, 𝒪_{ℙ(ℰ)}(2H − 3R)) the 4 Pfaffians of the matrix Ψ that involve the first column and consider the iterated mapping cone

[[𝒞^{5} → (𝒞^{3})^{⊕4}] → 𝒞^{0}] ,

where ∑_{i=1}^{4} 𝒞^{3} → 𝒞^{0} is induced by the multiplication with (Pf_{1}, …, Pf_{4}). This complex is a resolution of the ideal J generated by the 4 Pfaffians as an 𝒪_{ℙ^{12}}-module. In particular the minimized resolution is a subcomplex of the minimal free resolution of S_{C}. Since all extra syzygies are induced by the first column of Ψ, it follows that the 6^{th} syzygy modules in the linear strand of these minimal resolutions are canonically isomorphic. Therefore Syz_{6}(V (J)) and Syz_{6}(C) coincide and V (J) ⊂ Syz_{6}(C).

Proposition 4.5. Let C ⊂ ℙ^{12} be a general 5-gonal canonical curve. Then Syz_{6}(C) is the scheme cut out by the 4 Pfaffians of Ψ involving the first column. In particular Syz_{6}(C) = C ∪ p for some point p ∈ X.

Proof. One inclusion follows from the discussion above. The other inclusion follows by computing one example using Macaulay2. □

Remark 4.6. Since we do not have a full description of the space of extra syzygies for 5-gonal canonical curves of higher genus, a similar approach does not work in these cases.

Acknowledgement. This article is based on my master’s thesis. First of all I would like to thank my advisor Frank-Olaf Schreyer for valuable discussions on the topic and continuous support. I would further like to thank Michael Hoff and the referee for carefully reading this article and pointing out many typos and misprints.

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Christian Bopp

Mathematik und Informatik Universität des Saarlandes, Campus E2.4, 66123 Saarbrücken, Germany bopp@math.uni-sb.de 1070

Documenta Mathematica 20 (2015)