Syzygies of Prym and paracanonical curves of genus 8
Elisabetta Colombo, Gavril Farkas, Alessandro Verra, Claire Voisin
´´Epijournal de G´eom´etrie Alg´ebrique epiga.episciences.org
Volume 1 (2017), Article Nr. 7
Syzygies of Prym and paracanonical curves of genus Elisabetta Colombo, Gavril Farkas, Alessandro Verra, and Claire Voisin
Abstract.
By analogy with Green’s Conjecture on syzygies of canonical curves, the Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve ofgenus g is natural. The Prym-Green Conjecture is known to hold in odd genus for almost alllevels. Probabilistic arguments strongly suggested that the conjecture might fail for level 2and genus 8 or 16. In this paper, we present three geometric proofs of the surprising failureof the Prym-Green Conjecture in genus 8, hoping that the methods introduced here will shedlight on all the exceptions to the Prym-Green Conjecture for genera with high divisibilityby 2. Keywords.
Paracanonical curve; syzygy; genus 8; moduli of Prym varieties [Fran¸cais]Titre. Syzygies de Prym et courbes paracanoniques de genre 8R´esum´e.
Par analogie avec la conjecture de Green sur les syzygies des courbes canoniques,la conjecture de Prym-Green pr´edit que la r´esolution d’une courbe g´en´erale, paracanonique,de genre g et de niveau p est naturelle. Cette conjecture est connue en genre impair pourpresque tout niveau. Des arguments probabilistes ont fortement sugg´er´e qu’elle pourraits’av´erer fausse pour le niveau 2 en genre 8 et 16. Dans cet article, nous pr´esentons troisd´emonstrations g´eom´etriques de la surprenante non-validit´e de la conjecture de Prym-Greenen genre 8, en esp´erant que les m´ethodes introduites apporteront un ´eclairage nouveau surtoutes les exceptions `a la conjecture de Prym-Green pour des genres divisibles par une grandepuissance de 2. Received by the Editors on December 15, 2016, and in final form on June 23, 2017.Accepted on June 27, 2017.Elisabetta ColomboUniversit`a di Milano, Dipartimento di Matematica, Via Cesare Saldini 50, 20133 Milano, Italy e-mail : [email protected] FarkasHumboldt-Universit¨at zu Berlin, Institut f¨ur Mathematik, Unter den Linden 6, 10099 Berlin, Germany e-mail : [email protected] VerraUniversit`a Roma Tre, Dipartimento di Matematica, Largo San Leonardo Murialdo, 1-00146 Roma, Italy e-mail : [email protected] VoisinColl´ege de France 3, Rue d’Ulm, 75005 Paris, France e-mail : [email protected] © by the author(s) This work is licensed under http://creativecommons.org/licenses/by-sa/4.0/ a r X i v : . [ m a t h . AG ] J u l
1. Introduction
1. Introduction
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22. Syzygies of paracanonical curves of genus 8 . . . . . . . . . . . . . . . . . . . . . . . .
43. First proof: reducible spin curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R g . . . . . . . . . . . . . . . . . . . . . . vector bundles and singular quartics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction
By analogy with Green’s Conjecture on the syzygies of a general canonical curve [18], [19], the Prym-Green Conjecture, formulated in [10] and [3], predicts that the resolution of a paracanonical curve φ K C ⊗ η : C (cid:44) → P g − , where C is a general curve of genus g and η ∈ Pic ( C )[ (cid:96) ] is an (cid:96) -torsion point is natural. For evengenus g = 2 i + 6, the Prym-Green Conjecture amounts to the vanishing statement K i, ( C, K C ⊗ η ) = K i +1 , ( C, K C ⊗ η ) = 0 , (1.1)in terms of Koszul cohomology groups. Equivalently, the genus g paracanonical level (cid:96) curve C ⊆ P g − satisfies the Green-Lazarsfeld property ( N i ). The Prym-Green Conjecture has been proved for all odd genera g when (cid:96) = 2, see [8], or (cid:96) ≥ (cid:113) g +22 , see [9]. For even genus, the Prym-Green Conjecturehas been established by degeneration and using computer algebra tools in [3] and [4], for all (cid:96) ≤ g ≤
18, with two possible mysterious exceptions in level 2 and genus g = 8 ,
16 respectively. Thelast section of [3] provides various pieces of evidence, including a probabilistic argument, stronglysuggesting that for g = 8, one has dim K , ( C, K C ⊗ η ) = 1, and thus the vanishing (1.1) fails in thiscase. It is tempting to believe that the exceptions g = 8 ,
16 can be extrapolated to higher genus, andthat for genera g with high divisibility by 2, there are genuinely novel ways of constructing syzygies ofPrym-canonical curves waiting to be discovered. It would be very interesting to test experimentallythe next relevant case g = 24. Unfortunately, due to memory and running time constraints, this iscurrently completely out of reach, see [3] and [7].The aim of this paper is to confirm the expectation formulated in [3] and offer several geometricexplanations for the surprising failure of the Prym-Green Conjecture in genus 8, hoping that thegeometric methods described here for constructing syzygies of Prym-canonical curves will eventuallyshed light on all the exceptions to the Prym-Green Conjecture. We choose a general Prym-canonicalcurve of genus 8 φ K C ⊗ η : C (cid:44) → P , with η ⊗ = O C . Set L := K C ⊗ η and denote I C,L ( k ) := Ker (cid:8) Sym k H ( C, L ) → H ( C, L ⊗ k ) (cid:9) forall k ≥
2. Observe that dim I C,L (2) = dim K , ( C, L ) = 7 and dim I C,L (3) = 49, therefore as [
C, η ]varies in moduli, the multiplication map µ C,L : I C,L (2) ⊗ H ( C, L ) → I C,L (3)globalizes to a morphism of vector bundles of the same rank over the stack R classifying pairs [ C, η ],where C is a smooth curve of genus 8 and η ∈ Pic [2] \ {O C } . . Colombo, G. Farkas, A. Verra & C. Voisin, Syzygies of Prym and paracanonical curves of genus Theorem 1.
For a general Prym curve [ C, η ] ∈ R , one has K , ( C, L ) (cid:54) = 0 . Equivalently themultiplication map µ C,L : I C,L (2) ⊗ H ( C, L ) → I C,L (3) is not an isomorphism.
We present three different proofs of Theorem 1. The first proof, presented in Section 3 usesthe structure theorem already pointed out in [3] for degenerate syzygies of paracanonical curves in P . Precisely, if a paracanonical genus 8 curve φ K C ⊗ η : C (cid:44) → P , where η (cid:54) = O C , has a syzygy0 (cid:54) = γ ∈ K , ( C, K C ⊗ η ) of sub-maximal rank (see Section 2 for a precise definition), then the syzygyscheme of γ consists of an isolated point p ∈ P \ C and a residual septic elliptic curve E ⊆ P meeting C transversally along a divisor e of degree 14, such that if e is viewed as a divisor on C and E respectively, then e C ∈ | K C ⊗ η ⊗ | and e E ∈ |O E (2) | . (1.2)The union D := C ∪ E (cid:44) → P , endowed with the line bundle O D (1) is a degenerate spin curveof genus 22 in the sense of [5]. The locus of stable spin structures with at least 7 sections defines asubvariety of codimension 21 = (cid:0) (cid:1) inside the moduli space S − of stable odd spin curves of genus22. By restricting this condition to the locus of spin structures having D := C ∪ e E as underlyingcurve, it turns out that one has enough parameters to realize this condition for a general C ⊆ P ifand only if dim | K C ⊗ η ⊗ | = 7 , which happens precisely when η ⊗ ∼ = O C . Therefore for each Prym-canonical curve C ⊆ P of genus8 there exists a corresponding elliptic curve E ⊆ P such that the intersection divisor E · C verifies(1.2), which forces K , ( C, K C ⊗ η ) (cid:54) = 0.The second and the third proofs involve the reformulation given in Section 2.B (see Proposition5) of the condition that a paracanonical curve φ L : C (cid:44) → P have a non-trivial syzygy. Precisely, if φ L ( C ) is scheme-theoretically generated by quadrics, then K , ( C, L ) (cid:54) = 0, if and only if there existsa quartic hypersurface in P singular along C ⊆ P , which is not a quadratic polynomial in quadricsvanishing along C , that is, it does not belong to the image of the multiplication mapSym I C,L (2) → I C,L (4) . Equivalently, one has H ( P , I C/ P (4)) (cid:54) = 0.The second proof presented in Section 4 uses intersection theory on the stack R . The virtualKoszul divisor of Prym curves [ C, η ] ∈ R having K , ( C, K C ⊗ η ) (cid:54) = 0, splits into two divisors D and D respectively, corresponding to the case whether C ⊆ P is not scheme-theoretically cut outby quadrics, or H ( P , I C/ P (4)) (cid:54) = 0 respectively. We determine the virtual classes of both closures D and D . Using an explicit uniruled parametrization of R constructed in [11], we concludethat the class [ D ] ∈ CH ( R ) cannot possibly be effective (see Theorem 20). Therefore, again K , ( C, K C ⊗ η ) (cid:54) = 0, for every Prym curve [ C, η ] ∈ R .The third proof given in Section 5 even though subject to a plausible, but still unproved transver-sality assumption, is constructive and potentially the most useful, for we feel it might offer hints tothe case g = 16 and further. The idea is to consider rank 2 vector bundles E on C with canonicaldeterminant and h ( C, E ) = h ( C, E ( η )) = 4. (Note that the condition that η is 2-torsion is equiv-alent to the fact that E ( η ) also has canonical determinant, which is essential for the existence ofsuch nonsplit vector bundles, cf. [15].) By pulling back to C the determinantal quartic hypersurfaceconsisting of rank 3 tensors in P (cid:16) H ( C, E ) ∨ ⊗ H ( C, E ( η )) ∨ (cid:17) ∼ = P under the natural map H ( C, K C ⊗ η ) ∨ → H ( C, E ) ∨ ⊗ H ( C, E ( η )) ∨ , we obtain explicit quartichypersurfaces singular along the curve C ⊆ P . Our proof that these are not quadratic polynomials
2. Syzygies of paracanonical curves of genus 8
2. Syzygies of paracanonical curves of genus 8 into quadrics vanishing along the curve, that is, they do not lie in the image of Sym I C,L (2) remainsincomplete, but there is a lot of evidence for this.The methods of Section 5 suggests the following analogy in the next case g = 16. If [ C, η ] ∈ R is a Prym curve of genus 16, there exist vector bundles E on C with det E ∼ = K C and satisfying h ( C, E ) = h ( C, E ( η )) = 6. Potentially they could be used to prove that K , ( C, K C ⊗ η ) (cid:54) = 0 andthus confirm the next exception to the Prym-Green Conjecture.
2. Syzygies of paracanonical curves of genus 8
Let C be a general smooth projective curve of genus 8. For a non-trivial line bundle η ∈ Pic ( C ),we shall study the paracanonical line bundle L := K C ⊗ η . When η is a 2-torsion point, we speak ofthe Prym-canonical line bundle L . For each paracanonical bundle L , we have h ( C, L ) = 7 and aninduced embedding φ L : C (cid:44) → P . The goal is to understand the reasons for the non-vanishing of the Koszul group K , ( C, L ) of aPrym-canonical bundle L , as suggested experimentally by the results of [3], [4].Let I C (2) = I C,L (2) ⊆ H ( P , O P (2)), respectively I C (3) = I C,L (3) ⊆ H ( P , O P (3)) be theideal of quadrics, respectively cubics, vanishing on φ L ( C ). It is well-known that whenever L isprojectively normal, the non-vanishing of the Koszul cohomology group K , ( C, L ) is equivalent tothe non-surjectivity of the multiplication map µ C,L : H ( P , O P (1)) ⊗ I C (2) → I C (3) . (2.3)Note that dim I C (2) = (cid:18) (cid:19) −
21 = 7 , and dim I C (3) = (cid:18) (cid:19) − ·
14 + 7 = 49 , respectively, so that the two spaces appearing in the map (2.3) have the same dimension. Denote by P the universal degree 14 Picard variety over M consisting of pairs [ C, L ], where [ C ] ∈ M and L (cid:54) = K C . The jumping locus Kosz := (cid:110) [ C, L ] ∈ P : K , ( C, L ) (cid:54) = 0 (cid:111) is a divisor. It turns out, cf. Theorem 5.3 of [3] and Proposition 8, that Kosz splits into two componentsdepending on the rank of the corresponding non-zero syzygy from K , ( C, L ). Definition 2.
The rank of a non-zero syzygy γ = (cid:80) i =0 (cid:96) i ⊗ q i ∈ Ker( µ C,L ) is the dimension of thesubspace (cid:104) (cid:96) , . . . , (cid:96) (cid:105) ⊆ H ( P , O P (1)). The syzygy scheme Syz( γ ) of γ is the largest subscheme Y ⊆ P such that γ ∈ H ( P , O P (1)) ⊗ I Y (2).It is shown in [3], that Kosz splits into divisors
Kosz and Kosz , depending on whether the syzygy0 (cid:54) = γ ∈ Ker( µ C,L ) has rank 6 or 7 respectively. By a specialization argument to irreducible nodalcurves, it follows from [3] that R (cid:42) Kosz . A direct, more transparent proof of this fact will be givenin Proposition 13. with special syzygies and elliptic curves We summarize a few facts already stated or recalled in Section 5 of [3] concerning rank 6 syzygies ofparacanonical curves in P . Very generally, let γ = (cid:88) i =1 (cid:96) i ⊗ q i ∈ H ( P , O P (1)) ⊗ H ( P , O P (2)) . Colombo, G. Farkas, A. Verra & C. Voisin, Syzygies of Prym and paracanonical curves of genus be a rank 6 linear syzygy among quadrics in P . The linear forms (cid:96) , . . . , (cid:96) define a point p ∈ P .Following Lemma 6.3 of [16], there exists a skew-symmetric matrix of linear forms A := ( a ij ) i,j =1 ,..., ,such that q i = (cid:88) j =1 (cid:96) j a ij . In the space P with coordinates (cid:96) , . . . , (cid:96) and a ij for 1 ≤ i < j ≤
6, one considers the 15-dimensionalvariety X defined by the 6 quadratic equations (cid:80) j =1 (cid:96) j a ij = 0, where i = 1 , . . . , A ) = 0 in the variables a ij . The original space P embeds in P via evaluation. Thesyzygy scheme Syz( γ ) is the union of the point p and of the intersection D of P with the variety X .It follows from Theorem 4.4 of [6], that for a general rank 6 syzygy γ as above, D ⊆ P is a smoothcurve of genus 22 and degree 21 such that O D (1) is a theta characteristic.In the case at hand, that is, when [ C, L ] ∈ Kosz , the curve D must be reducible, for it has C asa component. More precisely: Lemma 3.
For a general paracanonical curve C ⊆ P having a rank syzygy, the curve D is nodaland consists of two components C ∪ E , where E ⊆ P is an elliptic septic curve. Furthermore, O D (2) = ω D . The intersection e := C · E , viewed as a divisor on C satisfies e C ∈ |O C (2) ⊗ K ∨ C | , andas a divisor on E , satisfies e E ∈ |O E (2) | . Remark 4.
Note that C is Prym-canonical or canonical if and only if e C ∈ | K C | .The construction above is reversible. Firstly, general element [ C, L ] ∈ Kosz can be reconstructedas the residual curve of a reducible spin curve D ⊆ P of genus 22 containing an elliptic curve E ⊆ P with deg( E ) = 7 as a component such that the union of D and some point p ∈ P \ E is the syzygyscheme of a rank 6 linear syzygy among quadrics in P .Furthermore, given a reducible spin curve D = C ∪ e E ⊆ P of genus 22 as above, that is, with ω D ∼ = O D (2), the genus 8 component C has a nontrivial syzygy of rank 6 involving the quadrics inthe 6-dimensional subspace I D (2) ⊆ I C (2), see Lemma 29 for a proof of this fact. We first discuss an alternative characterization of the non-surjectivity of the map µ C,L : Proposition 5.
Assume the paracanonical curve φ L ( C ) is projectively normal and scheme-theoreticallycut out by quadrics. Then K , ( C, L ) (cid:54) = 0 if and only if there exists a degree homogeneous poly-nomial on P , which vanishes to order at least along C but does not belong to the image of themultiplication map Sym I C,L (2) → I C,L (4) .Proof.
We work on the variety X τ → P defined as the blow-up of P along φ L ( C ). Let E be theexceptional divisor of the blow-up, and consider the line bundle H := τ ∗ O P (2)( − E ) on X . Its spaceof sections identifies to I C (2), and our assumption that C is scheme-theoretically cut out by quadricssays equivalently that H is a globally generated line bundle on X . The nonvanishing of K , ( C, L ) isequivalent to the non-surjectivity of the multiplication map H ( X, H ) ⊗ H ( X, τ ∗ O (1)) → H ( X, H ⊗ τ ∗ O (1)) , (2.4)where we use the identification H (cid:0) X, H ⊗ τ ∗ O (1) (cid:1) = H (cid:0) X, τ ∗ O (3)( − E ) (cid:1) = I C (3) .
2. Syzygies of paracanonical curves of genus 8
2. Syzygies of paracanonical curves of genus 8 As H is globally generated by its space W := I C (2) of global sections, the Koszul complex0 → (cid:94) W ⊗ O X ( − H ) → . . . → (cid:94) W ⊗ O X ( − H ) → W ⊗ O X ( − H ) → O X → τ ∗ O P (1)( H ) and take global sections. The last map is thenthe multiplication map (2.4). The successive terms of this twisted complex are i (cid:94) W ⊗ O X (cid:0) τ ∗ O (1) (cid:1) (( − i + 1) H ) , for 0 ≤ i ≤
7. The spectral sequence abutting to the hypercohomology of this complex, that is 0, has E , = coker (cid:110) W ⊗ H ( X, τ ∗ O (1)) → H ( X, H ⊗ τ ∗ O (1)) (cid:111) (2.6)and the terms E i, − i − for i < − (cid:86) − i W ⊗ H − i − (cid:0) X, τ ∗ O (1)(( i + 1) H ) (cid:1) . Similarly, wehave E i, − i = − i (cid:94) W ⊗ H − i (cid:0) X, τ ∗ O (1)(( i + 1) H ) (cid:1) . Lemma 6. (i)
We have E i, − i − = − i (cid:94) W ⊗ H − i − (cid:0) X, τ ∗ O (1)(( i + 1) H ) (cid:1) = 0 , (2.7) for − i − , . . . , . (ii) For − i − , that is, i = − , we have E − , = (cid:94) W ⊗ H (cid:0) X, τ ∗ O (1)( − H ) (cid:1) = (cid:94) W ⊗ I C (4) ∨ , (2.8) where I C (4) ⊆ I C (4) is the set of quartic polynomials vanishing at order at least along C ,and E − , = (cid:94) W ⊗ H (cid:0) X, τ ∗ O (1)( − H ) (cid:1) = (cid:94) W ⊗ I C (2) ∨ . (2.9)(iii) We have E i, − i = 0 , for − < i < .Proof of Lemma 6. (i) We want equivalently to show that H (cid:96) ( X, τ ∗ O (1)( − (cid:96)H )) = 0 , when (cid:96) = 5 , . . . , . Recall that H = τ ∗ O (2)( − E ). Furthermore, K X = τ ∗ O P ( − E ) . (2.10)So we have to prove that H (cid:96) (cid:0) X, τ ∗ O ( − (cid:96) + 1)( (cid:96)E ) (cid:1) = 0 , for (cid:96) = 5 , . . . , . (2.11)Examining the spectral sequence induced by τ , and using the fact that R s τ ∗ ( O X ( tE )) = 0 . Colombo, G. Farkas, A. Verra & C. Voisin, Syzygies of Prym and paracanonical curves of genus for s (cid:54) = 0 , s = 4 , t ≤
4, we see that for 1 ≤ (cid:96) ≤ H (cid:96) (cid:0) X, τ ∗ O ( − (cid:96) + 1)( (cid:96)E ) (cid:1) = H (cid:96) (cid:0) P , O ( − (cid:96) + 1) ⊗ R τ ∗ O X ( (cid:96)E ) (cid:1) . For 1 ≤ (cid:96) ≤
4, the right hand side is zero, because it is equal to H (cid:96) (cid:0) P , O ( − (cid:96) + 1) (cid:1) .For (cid:96) = 5, we have to compute the space H ( X, τ ∗ O ( − E )), which by Serre duality and by(2.10), is dual to the space H ( X, τ ∗ O (2)( − E )) = H ( P , O (2) ⊗ I C ) = 0 . (ii) We have to compute the spaces H ( X, τ ∗ O (1)( − H )) and H ( X, τ ∗ O (1)( − H )). As H := τ ∗ O (2)( − E ), this is rewritten as H ( X, τ ∗ O ( − E )) and H ( X, τ ∗ O ( − E )) respectively. If wedualize using (2.10), we get H (cid:0) X, τ ∗ O ( − E ) (cid:1) ∨ = H (cid:0) X, τ ∗ O (4)( − E ) (cid:1) = I C (4) ,H (cid:0) X, τ ∗ O ( − E ) (cid:1) ∨ = H (cid:0) X, τ ∗ O (2)( − E ) (cid:1) = I C (2) . (iii) We have E i, − i = E − , = − i (cid:94) W ⊗ H − i (cid:0) X, τ ∗ O (1)(( i + 1) H ) (cid:1) = − i (cid:94) W ⊗ H − i (cid:0) X, τ ∗ O (2 i + 3)(( − i − E ) (cid:1) . For 1 ≤ − i ≤
5, we have R s τ ∗ O X (cid:0) ( − i − E (cid:1) = 0 unless s = 0. Furthermore, we have R τ ∗ O X (cid:0) ( − i − E (cid:1) = O P , so that H − i (cid:0) X, τ ∗ O (2 i + 3)(( − i − E ) (cid:1) = H − i (cid:0) P , O P (2 i + 3) (cid:1) = 0 . (cid:3) Corollary 7.
Only one E p,q -terms of this spectral sequence is possibly nonzero in degree − , namely E − , = Ker (cid:110) (cid:94) W ⊗ I C (4) ∨ → (cid:94) W ⊗ I C (2) ∨ (cid:111) . (2.12) Furthermore, all the differentials d r starting from E − , vanish for ≤ r < . Note that the map (cid:94) W ⊗ I C (4) ∨ → (cid:94) W ⊗ I C (2) ∨ is nothing but the transpose of the multiplication map W ⊗ I C (2) → I C (4) , up to trivialization of (cid:86) W . It follows that( E − , ) ∨ = Coker (cid:110) W ⊗ I C (2) → I C (4) (cid:111) . (2.13)Corollary 7 concludes the proof of the proposition since it implies that we have an isomorphism givenby d between (2.12) and (2.6), or a perfect duality between (2.12) and the cokernel (2.13). (cid:3) Proposition 5 has the following consequence. Recall that P is the moduli space of pairs [ C, L ],with C being a smooth curve of genus 8 and L (cid:54) = K C a paracanonical line bundle.
2. Syzygies of paracanonical curves of genus 8
2. Syzygies of paracanonical curves of genus 8
Proposition 8.
The Koszul divisor
Kosz of P is the union of two divisors, one of them being theset of pairs [ C, L ] such that φ L ( C ) is not scheme-theoretically cut out by quadrics, the other being theset of pairs [ C, L ] such that H ( P , I C (4)) (cid:54) = 0 , or equivalently, such that there exists a quartic whichis singular along φ L ( C ) but does not lie in Sym I C (2) .Proof. We first have to prove that the locus of pairs [
C, L ] such that φ L ( C ) is not scheme-theoreticallycut-out by quadrics is contained in the divisor Kosz . This is a consequence of the following lemmas:
Lemma 9. If L (cid:54) = K C is a projectively normal paracanonical line bundle on a curve of genus , then φ L ( C ) is scheme-theoretically cut out by cubics.Proof of Lemma 9. We observe that the twisted ideal sheaf I C (3) is regular in Castelnuovo-Mumfordsense. Indeed, we have H i ( P , I C (3 − i )) = H i − ( C, L ⊗ (3 − i ) )for i ≥
2, and the right hand side is obviously 0 for i − ≥
2, and also 0 for i − H ( C, L ) = 0because L (cid:54) = K C and deg L = 2 g −
2. For i = 1, we have H ( P , I C (2)) = 0by projective normality. Being regular, the sheaf I C (3) is generated by global sections. (cid:3) Corollary 10. If C, L are as above, and C is not scheme-theoretically cut out by quadrics, then themultiplication map I C (2) ⊗ H ( P , O P (1)) → I C (3) is not surjective. To conclude the proof of the proposition, we just have to show that the sublocus of P where L is not projectively normal is not a divisor, since the statement of the proposition will be thenan immediate consequence of Proposition 5. We argue along the lines of [12]. First of all, a linebundle L of degree 14 is not generated by sections if and only if L = K C ( − x + y ) for some points x, y ∈ C . This determines a codimension 6 locus of P . Similarly L is not very ample if and only if L = K C ( − x − y + z + t ), for some points x, y, z, t of C , which is satisfied in a codimension 4 locusof P . Finally, assume L is very ample but φ L ( C ) is not projectively normal. EquivalentlySym H ( C, L ) → H ( C, L ⊗ )is not surjective, which means that there exists a rank 2 vector bundle F on C which is a nontrivialextension 0 −→ K C ⊗ L ∨ −→ F −→ L −→ , such that h ( C, F ) = 7. If x, y, z ∈ C , there exists a nonzero section σ ∈ H ( C, F ) vanishing on x, y and z , and thus F is also an extension0 −→ D −→ F −→ K C ⊗ D ∨ −→ , (2.14)where D is a line bundle such that h ( C, D ( − x − y − z )) (cid:54) = 0, and h ( C, L ⊗ D ∨ ) (cid:54) = 0. We thus have h ( C, D ) + h ( C, K C ⊗ D ∨ ) ≥ D ) ≤
2. As D is effective of degree at least 3, one has thefollowing possibilities:a) h ( C, K C ⊗ D ∨ ) = 0, and then D = L , which contradicts the fact that the extension (2.14) isnot split;b) h ( C, K C ⊗ D ∨ ) = 1 and h ( C, D ) ≥
6, and then D = L ( − x ) and h ( K C ⊗ L ∨ ( x )) (cid:54) = 0, so L = K C ( x − y ), which happens in a locus of codimension at least 6 in P ;c) D contributes to the Clifford index of C . As the locus of curves [ C ] ∈ M with Cliff( C ) ≤ M , this situation does not occur in codimension 1. (cid:3) . Colombo, G. Farkas, A. Verra & C. Voisin, Syzygies of Prym and paracanonical curves of genus We shall need later on the following result:
Lemma 11.
Let φ L : C (cid:44) → P be a projectively normal paracanonical curve of genus . If C isscheme-theoretically cut out by quadrics, the multiplication map Sym I C,L (2) → I C,L (4) (2.15) is injective.Proof.
As the restriction map φ ∗ L : H ( P , O P (2)) → H ( C, L ⊗ ) is surjective, its kernel I C,L (2) is ofdimension 7. Let as before τ : X → P be the blow-up of P along φ L ( C ), and let E be its exceptionaldivisor. We view I C,L (2) as H ( X, τ ∗ O (2)( − E )) and our assumption is that I C,L (2) generates theline bundle H := τ ∗ O (2)( − E ) everywhere on X . Thus I C,L (2) provides a morphism ψ : X → P ( I C,L (2)) . (2.16)Now we have deg c ( H ) (cid:54) = 0 by Sublemma 12 below, and thus the morphism ψ has to be genericallyfinite, hence dominant since both spaces have dimension 6. It is then clear that the pull-back map ψ ∗ : H (cid:0) P ( I C,L (2)) , O (2) (cid:1) → H ( X, H ⊗ )is injective. On the other hand, this morphism is nothing but the map (2.15). (cid:3) Sublemma 12.
With the same notation as above, we have deg c ( H ) = 8 . (2.17) Proof.
We have c ( H ) = (cid:88) i (cid:18) i (cid:19) ( − i h i · E − i , where h := τ ∗ c ( O P (1)), and h i · E − i = 0for i (cid:54) = 6 , ,
0. Furthermore h = 1 , and h · E = deg φ L ( C ) = 14and E = c ( N C ). By adjunction formuladeg c ( N C ) = 7deg φ L ( C ) + deg K C = 8 · . It follows that deg c ( H ) = 64 − ·
28 + 8 ·
14 = 8 , which proves (2.17). (cid:3) Proposition 5 and Lemma 11 describe precisely the splitting of the Koszul divisor
Kosz into thedivisors
Kosz and Kosz corresponding to paracanonical curves [ C, L ] ∈ P having a non-zero syzygy γ ∈ K , ( C, L ) of rank 6 or respectively 7. Precisely,
Kosz is a unirational divisor (cf. [3] Theorem5.3) consisting of those paracanonical curves C ⊆ P for which H ( P , I C (4)) (cid:54) = 0. The divisor Kosz consists of paracanonical curves C ⊆ P which are not scheme-theoretically cut out by quadrics.
3. First proof: reducible spin curves
3. First proof: reducible spin curves
3. First proof: reducible spin curves
The first observation is the following result (already observed experimentally in [3]), which turnsout to be useful for the description given below of the general paracanonical curve of genus 8 withnontrivial syzygies.
Proposition 13.
Let C ⊆ P be a smooth paracanonical curve of genus and degree , scheme-theoretically generated by quadrics. Then a nontrivial syzygy γ ∈ Ker (cid:8) I C (2) ⊗ H ( O P (1)) → I C (3) (cid:9) must be degenerate, that is of rank at most .Proof. We use the morphism ψ : X → P ( I C (2))introduced in (2.16), where τ : X → P is the blow-up of C with exceptional divisor E , and H := τ ∗ O P ( − E ). This gives us a morphism( τ, ψ ) : X → P × P which is of degree 1 on its image, and the syzygy γ induces a hypersurface Y of bidegree (1 ,
1) in P × P containing the 6-dimensional variety ( τ, ψ )( X ). Assume to the contrary that γ has maximalrank 7, or equivalently that Y is smooth. Then by the Lefschetz Hyperplane Restriction Theorem,the restriction map H ( P × P , Z ) → H ( Y, Z ) is surjective, so that [( τ, ψ )( X )] Y ∈ H ( Y, Z ) isthe restriction of a class β ∈ H ( P × P , Z ), which implies that[( τ, ψ )( X )] = β · [ Y ] in H ( P × P , Z ) , (3.18)where [ Y ] ∈ H ( P × P , Z ) is the class of Y , that is h + h , with h i for i = 1 , H ( P × P , Z ) is the set of degree 6 homogeneouspolynomials with integral coefficients in h and h . We now have: Lemma 14.
An element α ∈ H ( P × P , Z ) is of the form ( h + h ) · β if and only if it satisfiesthe condition (cid:88) i =0 ( − i h i · h − i · α = 0 in H ( P × P , Z ) = Z . (3.19) Proof of Lemma 14.
We have ( h + h ) · (cid:0)(cid:80) i ( − i h i · h − i (cid:1) = 0 in H ( P × P , Z ), so one implicationis obvious. That the two conditions are equivalent then follows from the fact that both conditionsdetermine a saturated corank 1 sublattice of H ( P × P , Z ). (cid:3) To conclude that γ has to be degenerate, in view of Lemma 14, it suffices to prove that the class[( τ, ψ )( X )] does not satisfy (3.19). Since ( τ, ψ ) ∗ h = c ( H ) and ( τ, ψ ) ∗ h = 2 c ( H ) − E , it is enoughto prove that (cid:88) i =0 ( − i c ( H ) i · (2 c ( H ) − E ) − i (cid:54) = 0 , which follows from the computations made in the proof of Sublemma 12. (cid:3) . Colombo, G. Farkas, A. Verra & C. Voisin, Syzygies of Prym and paracanonical curves of genus Recall that S − g denotes the moduli stack of odd stable spin curves of genus g , see [5] for details. Westart with a nodal genus 22 spin curve of the form [ D := C ∪ E, ϑ ] ∈ S − , where C is a smooth genus8 curve, E is a smooth elliptic curve and e := C ∩ E consists of 14 distinct points, thus p a ( D ) = 22.Assume ϑ ∈ Pic ( D ) verifies ϑ ⊗ ∼ = ω D , hence the restricted line bundles ϑ E and ϑ C have degrees7 and 14 respectively. Furthermore, h ( E, ϑ E ) = 7, whereas h ( C, ϑ C ) = 7 if and only if ϑ C (cid:29) K C .The intersection divisor e on the two components of D is characterized by e C ∈ | ϑ ⊗ C ⊗ K ∨ C | and e E ∈ | ϑ ⊗ E | . Note in particular that e C ∈ | K C | if and only if ϑ ⊗ C = K ⊗ C , that is ( C, ϑ C ) is canonical or Prymcanonical.The line bundle ϑ on D fits into the Mayer-Vietoris exact sequence:0 −→ ϑ −→ ϑ C ⊕ ϑ E r −→ O e ( ϑ ) −→ , where r is defined by the isomorphisms on the fibers of ϑ C and ϑ E over the points in e . Given ϑ C ∈ Pic ( C ) with ϑ ⊗ C = K C ( e ) and ϑ E ∈ Pic ( E ) with ϑ ⊗ E = O E ( e ), there is a finite numberof stable spin curves [ D, θ ] ∈ S − such that the restrictions of ϑ to C and E are isomorphic to ϑ C and ϑ E respectively. Passing to global sections in the Mayer-Vietoris sequence, we obtain the exactsequence: 0 −→ H ( D, ϑ ) −→ H ( C, ϑ C ) ⊕ H ( E, ϑ E ) r −→ H ( O e ( ϑ )) −→ · · · . (3.20)Note that r is represented by a 14 ×
14 matrix and h ( D, ϑ ) = 14 − rk( r ). In the case of a reducible spincurve coming from the syzygy of a paracanonical genus 8 curve in Kosz , one has h ( D, ϑ ) = rk( r ) = 7. Theorem 1 states that every Prym canonical curve of genus 8 has a syzygy of rank 6. First we observethe existence of such a curve having the generic behavior described in Lemma 3.
Lemma 15.
There exists a curve [ C, η ] ∈ R , whose Prym canonical model is scheme theoreticallycut out by quadrics, and K , ( C, K C ⊗ η ) is -dimensional, generated by a syzygy γ of rank . Thesyzygy scheme of γ is the union of a point p and a nodal curve D = C ∪ E , such that E is a smoothelliptic curve of degree and e := C · E ∈ | K C | consists of mutually distinct points. Moreover, nocubic polynomial on P vanishes with multiplicity along C .Proof. Examples of singular
Prym canonical curves having all these properties have been producedin [3] Proposition 4.4 or [4]. A generic deformation in R of these singular examples will provide therequired smooth Prym canonical curve. (cid:3) (First) proof of Theorem 1. We denote by X the moduli space of elements [ C, η, x , . . . , x ], where[ C, η ] ∈ R is a smooth Prym curve of genus 8 and x i ∈ C are pairwise distinct points such that x + · · · + x ∈ | K C | ∼ = P . Since the fibres of the forgetful map X → R are 7-dimensional, it followsthat X is an irreducible variety of dimension 28.Let T be the locally closed parameter space of odd genus 22 spin curves having the form (cid:16)(cid:2) D := C ∪ { x ,...,x } E, ϑ (cid:3) : [ C ] ∈ M , (cid:88) i =1 x i ∈ | K C | , [ E, x , . . . , x ] ∈ M , , ϑ ⊗ = ω D (cid:17) .
4. Second proof: Divisor class calculations on R g
4. Second proof: Divisor class calculations on R g Observe that points in T , apart from the spin structure [ D, ϑ ] ∈ S − also carry an underlyingPrym structure [ C, η := K C ⊗ ϑ ∨ C ] ∈ R , for ϑ ⊗ C ∼ = K C ( x + · · · + x ) ∼ = K ⊗ C . One has an inducedfinite morphism T → X × M , , as well as a map µ : T → R forgetting the 14-pointed ellipticcurve. It follows that dim T = dim X + dim M , = 42. The locus T := (cid:8) [ D, ϑ ] ∈ T : h ( D, ϑ ) ≥ (cid:9) has the structure of a skew-symmetric degeneracy locus. Applying [13] Theorem 1.10, each componentof T has codimension at most (cid:0) (cid:1) = 21 inside T , that is, dim( T ) ≥ dim( R ).By passing to a general 8-nodal Prym canonical curve [ C, η ], following [3] Proposition 4.5, as wellas Lemma 15, we have that dim K , ( C, K C ⊗ η ) = 1. In particular, the fibre µ − ([ C, η ]) containsan isolated point, which shows that T is non-empty and has a component which maps dominantlyunder µ onto R . Theorem 1 now follows. (cid:3) Remark 16.
The same construction can be carried out at the level of general paracanonical curves[
C, L ] ∈ P , where L ∈ Pic ( C ) \ { K C } . The key difference is that we replace T by a variety T (cid:48) parametrizing objects (cid:16)(cid:2) D := C ∪ { x ,...,x } E, ϑ, L (cid:3) : [
C, x , . . . , x ] ∈ M , , L ∈ Pic ( C ) \ { K C } , (cid:88) i =1 x i ∈ | L ⊗ ⊗ K ∨ C | , [ E, x , . . . , x ] ∈ M , , ϑ ⊗ = ω D (cid:17) . Similarly, we have a morphism µ (cid:48) : T (cid:48) → P retaining the pair [ C, L ] alone. The main differencecompared to the Prym canonical case is that nowdim | L ⊗ ⊗ K ∨ C | = 6 , therefore dim( T (cid:48) ) = dim( P ) + dim( M , ) + 6 = 49. The degeneracy locus T (cid:48) ⊆ T (cid:48) defined by thecondition h ( D, ϑ ) ≥
7) has codimension 21 inside T (cid:48) , that is,dim( T (cid:48) ) = 28 = dim( P ) − . It follows that the image µ (cid:48) ( T (cid:48) ) ⊆ P has codimension 1, which is in accordance with Kosz being adivisor in P .
4. Second proof: Divisor class calculations on R g Recall [10] that R g is the Deligne-Mumford moduli space of Prym curves of genus g , whose geometricpoints are triples [ X, η, β ], where X is a quasi-stable curve of genus g , η ∈ Pic( X ) is a line bundle oftotal degree 0 such that η E = O E (1) for each smooth rational component E ⊆ X with | E ∩ X − E | = 2(such a component is said to be exceptional ), and β : η ⊗ → O X is a sheaf homomorphism whoserestriction to any non-exceptional component is an isomorphism. If π : R g → M g is the map droppingthe Prym structure, one has the formula π ∗ ( δ ) = δ (cid:48) + δ (cid:48)(cid:48) + 2 δ ram0 ∈ CH ( R g ) , (4.21)where δ (cid:48) := [∆ (cid:48) ] , δ (cid:48)(cid:48) := [∆ (cid:48)(cid:48) ], and δ ram0 := [∆ ram0 ] are irreducible boundary divisor classes on R g ,which we describe by specifying their respective general points.We choose a general point [ C xy ] ∈ ∆ ⊂ M g corresponding to a smooth 2-pointed curve ( C, x, y )of genus g − ν : C → C xy , where ν ( x ) = ν ( y ). A general point . Colombo, G. Farkas, A. Verra & C. Voisin, Syzygies of Prym and paracanonical curves of genus of ∆ (cid:48) (respectively of ∆ (cid:48)(cid:48) ) corresponds to a pair [ C xy , η ], where η ∈ Pic ( C xy )[2] and ν ∗ ( η ) ∈ Pic ( C )is non-trivial (respectively, ν ∗ ( η ) = O C ). A general point of ∆ ram0 is a Prym curve of the form ( X, η ),where X := C ∪ { x,y } P is a quasi-stable curve with p a ( X ) = g and η ∈ Pic ( X ) is a line bundlesuch that η P = O P (1) and η C = O C ( − x − y ). In this case, the choice of the homomorphism β isuniquely determined by X and η . In what follows, we work on the partial compactification (cid:101) R g ⊆ R g of R g obtained by removing the boundary components π − (∆ j ) for j = 1 , . . . , (cid:98) g (cid:99) , as well as ∆ (cid:48)(cid:48) . Inparticular, CH ( (cid:101) R g ) = Q (cid:104) λ, δ (cid:48) , δ ram0 (cid:105) .For a stable Prym curve [ X, η ] ∈ (cid:101) R g , set L := ω X ⊗ η ∈ Pic g − ( X ) to be the paracanonicalbundle. For i ≥
1, we introduce the vector bundle N k over (cid:101) R g , having fibres N k [ X, η ] = H ( X, L ⊗ k ) . The first Chern class of N k is computed in [10] Proposition 1.7: c ( N k ) = (cid:18) k (cid:19)(cid:16) λ − δ (cid:48) − δ ram0 (cid:17) + λ − k δ ram0 . (4.22)Then we define the locally free sheaves G k on (cid:101) R g via the exact sequences0 −→ G k −→ Sym k N −→ N k −→ , that is, satisfying G k [ X, η ] := I X,L ( k ) ⊆ Sym k H ( X, L ). Using (4.22) one computes c ( G k ).We also need the class of the vector bundle G with fibres G [ X, η ] = H ( X, ω ⊗ X ⊗ η ⊗ ) = H ( X, ω X ⊗ L ⊗ ) . Lemma 17.
One has c ( G ) = 121 λ − δ (cid:48) − δ ram0 ∈ CH ( (cid:101) R g ) .Proof. We apply Grothendieck-Riemann-Roch to the universal Prym curve f : C → (cid:101) R g . Denote by L ∈
Pic( C ) the universal Prym bundle , whose restriction to each Prym curve is the corresponding2-torsion point, that is, L | f − ([ X,η ]) = η , for each point [ X, η ] ∈ (cid:101) R g . Since R f ∗ ( ω ⊗ f ⊗ L ⊗ ) = 0, wewrite c ( G ) = f ∗ (cid:104)(cid:16) c ( ω f ) + 4 c ( L ) + (5 c ( ω f ) + 4 c ( L )) (cid:17) · (cid:16) − c ( ω f )2 + c (Ω f ) + [Sing( f )]12 (cid:17)(cid:105) . We use then the formulas f ∗ ( c ( L )) = − δ ram0 / f ∗ ( c ( L ) · c ( ω f )) = 0 (see [10], Proposition 1.6)coupled with Mumford’s formula f ∗ ( c (Ω f ) + [Sing( f )]) = 12 λ as well with the identity κ := f ∗ ( c ( ω f )) = 12 λ − δ (cid:48) − δ ram0 , in order to conclude. (cid:3) The Koszul locus Z := Kosz ∩ R = (cid:110) [ C, η ] ∈ R : K , ( C, K C ⊗ η ) (cid:54) = 0 (cid:111) is a virtual divisor on R , that is, the degeneracy locus of a map between vector bundles of the samerank over (cid:101) R . If it is a genuine divisor (which we aim to rule out), the class of its closure in (cid:101) R isgiven by [3] Theorem F: [ Z ] = 27 λ − δ (cid:48) − δ ram0 ∈ CH ( (cid:101) R ) .
4. Second proof: Divisor class calculations on R g
4. Second proof: Divisor class calculations on R g Remark 18.
Some of the considerations above can be extended to higher order torsion points. Werecall that R g,(cid:96) is the moduli space of pairs [ C, η ], where C is a smooth curve of genus g and η ∈ Pic ( C )is a non-trivial (cid:96) -torsion point. It is then shown in [3] that the locus Z ,(cid:96) := Kosz ∩ R ,(cid:96) ⊆ P is adivisor on R ,(cid:96) for each other level (cid:96) ≥
3. The class of the compactification of Z ,(cid:96) is given by thefollowing formula, see [3] Theorem F:[ Z ,(cid:96) ] = 27 λ − δ (cid:48) − (cid:98) (cid:96) (cid:99) (cid:88) a =1 a − a(cid:96) + (cid:96) ) (cid:96) δ ( a )0 ∈ CH ( (cid:101) R ,(cid:96) ) . We refer to [3] Section 1.4, for the definition of the boundary divisor classes δ ( a )0 , where a = 1 , . . . , (cid:98) (cid:96) (cid:99) .If π : R g,(cid:96) → M g is the map forgetting the level (cid:96) structure, then π ∗ ( δ ) = δ (cid:48) + δ (cid:48)(cid:48) + (cid:96) (cid:98) (cid:96) (cid:99) (cid:88) (cid:96) =1 δ ( a )0 . We fix now a genus 8 Prym-canonically embedded curve φ L : C (cid:44) → P . As usual, we denote thekernel bundle by M L := Ω P | C (1), hence we have the exact sequence0 −→ N ∨ C ⊗ L ⊗ −→ M L ⊗ L ⊗ −→ K C ⊗ L ⊗ −→ . (4.23)This can be interpreted as an exact sequence of vector bundles over (cid:101) R . Denoting by H the vectorbundle over (cid:101) R with fibres H ( C, N ∨ C ⊗ L ⊗ ), we compute using the previous formulas and the factthat rk( N ) = h ( C, L ) = 7 and rk( N ) = h ( C, L ⊗ ) = 35: c ( H ) = 35 c ( N ) + 7 c ( N ) − c ( N ) − c ( G ) = 100 λ − δ (cid:48) − δ ram0 . (4.24)Thus D = Kosz ∩ R and D = Kosz ∩ R . We have already seen in Proposition 5 that K , ( C, L ) (cid:54) = 0 if and only if either φ L ( C ) ⊆ P is not scheme-theoretically cut out by quadrics, orelse, H ( P , I C (4)) (cid:54) = 0. We write Z = D + D , where D := (cid:110) [ C, η ] ∈ R : φ L ( C ) ⊆ P is scheme-theoretically not cut out by quadrics (cid:111) and D := (cid:110) [ C, η ] ∈ R : H ( P , I C (4)) (cid:54) = 0 (cid:111) . We have already observed that dim I C,L (2) = 7 and χ ( P , I C (4)) = 28. If Z is a divisor, then D is a divisor as well and for [ C, η ] ∈ R \ D , we have thatdim Sym I C,L (2) = dim I C,L (4) = 28 . Paying some attention to its definition, the divisor D can be thought as the degeneracy locus (cid:110) [ C, η ] ∈ R : Sym I C,L (2) (cid:54) = −→ I C,L (4) (cid:111) , which is an effective divisor on (cid:101) R . We compute the class of this divisor: Theorem 19.
We have the following formulas: [ D ] = 7 λ − δ (cid:48) − δ ram0 ∈ CH ( (cid:101) R ) and [ D ] = 20 λ − δ (cid:48) − δ ram0 ∈ CH ( (cid:101) R ) . . Colombo, G. Farkas, A. Verra & C. Voisin, Syzygies of Prym and paracanonical curves of genus Proof.
We first globalize over (cid:101) R the following exact sequence:0 −→ I C,L (4) −→ I C,L (4) −→ H ( C, N ∨ C ⊗ L ⊗ ) −→ H ( P , I C (4)) −→ . Denote by A the sheaf on (cid:101) R supported along the divisor D , whose fibre over a general point of thatdivisor is equal to to H ( P , I C (4)). There is a surjective morphism of sheaves H → A and denote by G (cid:48) its kernel. Since A is locally free along D and (cid:101) R is a smooth stack, usingthe Auslander-Buchsbaum formula we find that G (cid:48) is a locally free sheaf of rank equal to rk( H ) = χ ( C, N ∨ C (4 L )) = 19 ·
7. Precisely, G (cid:48) is an elementary transformation of H along the divisor D .Furthermore, c ( G (cid:48) ) = c ( H ) − [ D ].The morphism G → H globalizing the maps I C,L (4) → H ( C, N ∨ C ⊗ L ⊗ ) factors through thesubsheaf G (cid:48) and we form the exact sequence:0 −→ G −→ G −→ G (cid:48) −→ . The multiplication maps Sym I C,L (2) → I C,L (4) globalize to a sheaf morphism ν : Sym ( G ) → G between locally free sheaves of the same rank 28 over the stack (cid:101) R . The degeneration locus of ν isprecisely the divisor D . We compute: c (Sym ( G )) = 8 c ( G ) = 8(8 c ( N ) − c ( N )) = − λ + 8( δ (cid:48) + δ ram0 ) , and c ( G ) = 120 c ( N ) − c ( N ) − c ( H ) + [ D ] = − λ + 11 δ (cid:48) + 252 δ ram0 + [ D ] . We obtain the relation [ D ] − [ D ] = − λ + 3 δ (cid:48) + δ ram0 . Since at the same time[ D ] + [ D ] = [ Z ] = 27 λ − δ (cid:48) − δ ram0 , we solve the system and conclude. (cid:3) We are now in a position to give a second proof of Theorem 1:
Theorem 20.
The class [ D ] cannot be effective. It follows that Z = R and K , ( C, K C ⊗ η ) (cid:54) = 0 ,for every Prym curve [ C, η ] ∈ R .Proof. We use the sweeping curve of the boundary divisor ∆ (cid:48) of (cid:101) R constructed via Nikulin surfacesin [11] Lemma 3.2: Precisely, through the general point of ∆ (cid:48) there passes a rational curve Γ ⊆ ∆ (cid:48) ,entirely contained in (cid:101) R , having the following numerical characters:Γ · λ = 8 , Γ · δ (cid:48) = 42 , and Γ · δ ram0 = 8 . We note that Γ · D <
0. Writing D ≡ α · δ (cid:48) + E , where α ≥ E is an effective divisor whosesupport is disjoint from ∆ (cid:48) , we immediately obtain a contradiction. (cid:3)
5. Rank 2 vector bundles and singular quartics
5. Rank 2 vector bundles and singular quartics
The divisors D and D can be defined in an identical manner at the level of each moduli space R ,(cid:96) of twisted level (cid:96) curves of genus g . As already pointed out, in the case (cid:96) ≥ D and D are actual divisors. Repeating the same calculations as for (cid:96) = 2, we obtain the following formula on the partial compactification (cid:101) R ,(cid:96) of R ,(cid:96) :[ D ] = 20 λ − δ (cid:48) − (cid:98) (cid:96) (cid:99) (cid:88) a =1 (cid:96) (7 a − a(cid:96) + 17 (cid:96) − (cid:96) ) δ ( a )0 ∈ CH ( (cid:101) R ,(cid:96) ) . (4.25)As an application, we mention a different proof of one of the main results from [1]: Theorem 21.
The canonical class of R ,(cid:96) is big for (cid:96) ≥ . It follows that R ,(cid:96) is a variety of generaltype for (cid:96) = 3 , , .Proof. Using formula (4.25), it is a routine exercise to check that for (cid:96) ≥ K (cid:101) R ,(cid:96) = 13 λ − δ (cid:48) − ( (cid:96) + 1) (cid:98) (cid:96) (cid:99) (cid:88) a =1 δ ( a )0 can be written as a positive combination of the big class λ and the effective class [ D ], hence it is big.Arguing along the lines of [3] Remark 3.5, it is easy to extend this result to the full compactification R ,(cid:96) and deduce that K R ,(cid:96) is big.To conclude that R ,(cid:96) is of general type, one needs, apart from the bigness of the canonical class K (cid:101) R ,(cid:96) of the moduli stack, a result that the singularities of the coarse moduli space R ,(cid:96) impose noadjunction conditions. This is only known for 2 ≤ (cid:96) ≤ , (cid:96) (cid:54) = 5, see [2]. (cid:3)
5. Rank vector bundles and singular quartics Our goal in this section is to propose a construction of syzygies of Prym canonical curves of genus 8.We also sketch the proof of the fact that these syzygies are nontrivial. We fix again a general element[
C, η ] ∈ R and set L := K C ⊗ η . According to Proposition 5, in order to prove that K , ( C, L ) (cid:54) = 0,we have to produce quartic hypersurfaces in P which vanish at order at least 2 along φ L ( C ), but donot lie in the image of the map Sym I C,L (2) → I C,L (4). The goal of this section is to produce suchquartics from rank 2 vector bundles on C . The (incomplete) proof that the quartics we construct arenot in the image of Sym I C,L (2) depends on an unproved general position statement ( ∗ ), but theremight be other approaches exploiting the fact that the hypersurfaces in question are determinantal.The following construction produces quartics vanishing at order 2 along C . Let E be a rank 2vector bundle on C , with determinant K C . Assume h ( C, E ) = 4 , h ( C, E ( η )) = 4 . (5.26)Setting V := H ( C, E ) and V := H ( C, E ( η )), we have a natural map V ⊗ V → H ( C, L ) , defined using evaluation and the following composite map: H ( E ) ⊗ H ( E ( η )) → H ( E ⊗ E ( η )) ∼ = H ( E nd E ⊗ L ) Tr −→ H ( C, L ) . (5.27)This map gives dually a morphism H ( C, L ) ∨ → V ∨ ⊗ V ∨ , (which will be proved below to be injective for a general choice of E ). We consider the quartichypersurface D on P ( V ∨ ⊗ V ∨ ) parametrizing tensors of rank at most 3. . Colombo, G. Farkas, A. Verra & C. Voisin, Syzygies of Prym and paracanonical curves of genus Lemma 22.
The restriction D ,E of this quartic to P (cid:0) H ( C, L ) ∨ (cid:1) ⊆ P (cid:0) V ∨ ⊗ V ∨ (cid:1) is singular alongthe curve C .Proof. The quartic D is singular along the set T ⊆ P ( V ∨ ⊗ V ∨ ) of tensors of rank at most 2. Thequartic D ,E in P ( H ( C, L ) ∨ ) is thus singular along T ∩ P ( H ( C, L ) ∨ ), which obviously contains C ⊆ P (cid:0) H ( C, L ) ∨ (cid:1) , since at a point p ∈ C , the map V ⊗ V → H ( C, L ) composed with theevaluation at p factors through E | p ⊗ E ( η ) | p . (cid:3) By Brill-Noether theory, the variety W ( C ) of degree 7 pencils on C is 4-dimensional. Thereshould thus exist finitely many elements D ∈ W ( C ) with the property that h ( C, D ) ≥ , h ( C, D ⊗ η ) ≥ . (5.28)We now have the following lemma: Lemma 23.
Let [ C, η ] ∈ R be as above and D ∈ W ( C ) satisfying (5.28). Then (i) h ( C, D ) = 2 and h ( C, D ⊗ η ) = 2 . The multiplication map (cid:16) H ( C, D ) ⊗ H ( C, K C ⊗ D ∨ ) (cid:17) ⊕ (cid:16) H ( C, D ⊗ η ) ⊗ H ( C, K C ⊗ D ∨ ⊗ η ) (cid:17) → H ( C, K C ) is surjective (in fact, an isomorphism). (ii) The multiplication map (cid:16) H ( C, D ) ⊗ H ( C, K C ⊗ D ∨ ⊗ η ) (cid:17) ⊕ (cid:16) H ( C, D ⊗ η ) ⊗ H ( C, K C ⊗ D ∨ ) (cid:17) → H ( C, K C ( η )) is surjective.Proof. This can be proved by a degeneration argument, for example by degenerating C to the unionof two curves of genus 4 meeting at one point. (cid:3) By Brill-Noether theory, the following corollary follows from (i) above:
Corollary 24.
For [ C, η ] as above, the set of pencils D ∈ W ( C ) satisfying (5.28) is finite. Given such a D , we form the rank 2 vector bundle E = D ⊕ ( K C ⊗ D ∨ )on C which satisfies the conditions (5.26). The associated quartic is however not interesting for ourpurpose, due to the following fact: Lemma 25.
The quartic on P ( H ( C, L ) ∨ ) associated to the vector bundle D ⊕ ( K C ⊗ D ∨ ) is theunion of the two quadrics Q and Q associated respectively with the multiplication maps H ( D ) ⊗ H (( K C ⊗ D ∨ )( η )) → H ( K C ( η )) and H ( D ( η )) ⊗ H ( K C ⊗ D ∨ ) → H ( K C ( η )) . Both these quadrics contain C .Proof. Indeed we have in this case V = H ( C, E ) = H ( C, D ) ⊕ H ( C, K C ⊗ D ∨ ) , respectively V = H ( C, E ( η )) = H ( C, D ⊗ η ) ⊕ H ( C, K C ⊗ D ∨ ⊗ η ) .
5. Rank 2 vector bundles and singular quartics
5. Rank 2 vector bundles and singular quartics
Furthermore, it is clear that the map of (5.27) factors through the projection V ⊗ V → (cid:16) H ( C, D ) ⊗ H (cid:0) C, K C ⊗ D ∨ ⊗ η (cid:1)(cid:17) ⊕ (cid:16) H ( C, K C ⊗ D ∨ ) ⊗ H ( C, D ⊗ η ) (cid:17) and induces on each summand the multiplication map. The quadric Q is by definition associatedwith the the multiplication map µ : H ( C, D ) ⊗ H ( C, K C ⊗ D ∨ ⊗ η ) → H ( C, K C ⊗ η ) , and is the set of elements f in P ( H ( K C ⊗ η )) ∨ such that µ ∗ ( f ) is a tensor of rank ≤
1. Similarlyfor Q , with D being replaced with D ( η ). Finally we use the fact that a tensor( µ ∗ f, µ ∗ f ) ∈ (cid:16) H ( C, D ) ⊗ H ( C, K C ⊗ D ∨ ⊗ η ) (cid:17) ⊕ (cid:16) H ( C, K C ⊗ D ∨ ) ⊗ H ( C, D ⊗ η ) (cid:17) has rank at most 3 if and only if one of µ ∗ f and µ ∗ f has rank at most 1. (cid:3) We recall from [15] or [14] that the Brill-Noether condition h ( C, E ) ≥ (cid:0) (cid:1) equations on the parameter space of rank 2 vector bundles E with determinant K C . As det E ( η ) ∼ = K C as well, we conclude that the equations (5.26) impose only 20 conditions. As the moduli space SU C (2 , K C ) of semistable rank 2 vector bundles on C having determinant K C has dimension 3 g − C satisfying (5.26).We now sketch the proof of the fact that for C general of genus 8 and D ∈ W ( C ) satisfying(5.28), for a general deformation E of the vector bundle D ⊕ ( K C ⊗ D ∨ ) satisfying det E ∼ = K C and h ( C, E ) = 4, the associated quartic D ,E singular along C is not defined by an element of Sym I C (2).Combined with Proposition 5, this provides a third approach to Theorem 1. The proof of this factrests on an unproven general position statement ( ∗ ), so it is incomplete. Sketch of proof of the nontriviality of the syzygy.
The vector bundle E is generated by sections, as itis a general section-preserving deformation of the vector bundle D ⊕ ( K C ⊗ D ∨ )which is generated by global sections, and similarly for E ( η ). Along C ⊆ P ( H ( C, L ) ∨ ), then therational map P (cid:0) H ( C, L ) ∨ (cid:1) (cid:57)(cid:57)(cid:75) P (cid:0) H ( E ) ∨ ⊗ H ( E ( η )) ∨ (cid:1) is well-defined and the image of C is contained in the locus T ,E of tensors of rank exactly 2. Infact, the case of D ⊕ ( K C ⊗ D ∨ ) shows that this map is a morphism for general E (one just needsto know that H ( C, K C ⊗ η ) is generated by the two vector spaces H ( D ) ⊗ H ( K C ⊗ D ∨ ⊗ η ) and H ( D ⊗ η ) ⊗ H ( K C ⊗ D ∨ ) respectively, or rather their images under the multiplication map. Notethat on T ,E , there is a rank 2 vector bundle M which restricts to E on C .In the case of the split vector bundle E sp = D ⊕ ( K C ⊗ D ∨ ), Lemma 25 shows that the Zariskiclosure T ,E sp parameterizing tensors of rank ≤ P ( H ( C, L ) ∨ ) ⊆ P ( V ∨ ⊗ V ∨ ) is equal to thesingular locus of D ,E sp and consists of the union of the two planes P , P defined as the singular lociof the quadrics Q , Q respectively, and the intersection Q ∩ Q . The locus T ,E sp \ T ,E sp is the locuswhere the tensor has rank 1, and this happens exactly along the two conics P ∩ Q and P ∩ Q .The curve C is contained in Q ∩ Q and does not intersect P ∪ P . In particular, the rational map φ : (cid:102) P (cid:57)(cid:57)(cid:75) P given by the linear system I C (2) is well defined along P ∪ P . We believe that thefollowing general position statement concerning the two planes P i is true for general C and D, η asabove. . Colombo, G. Farkas, A. Verra & C. Voisin, Syzygies of Prym and paracanonical curves of genus ( ∗ ) The surfaces φ ( P i ) are projectively normal Veronese surfaces, generating a hyperplane (cid:104) φ ( P i ) (cid:105) ⊆ P . Furthermore, the surface φ ( P ) ∪ φ ( P ) ⊆ P is contained in a unique quadric in P , namely theunion of the two hyperplanes (cid:104) φ ( P ) (cid:105) and (cid:104) φ ( P ) (cid:105) . We now prove that, assuming ( ∗ ), for a general vector bundle E as above, the associated quartic D ,E singular along C is not defined by an element of Sym I C (2). As P , P are 2-dimensional reducedcomponents of T ,E sp , hence of the right dimension, the theory of determinantal hypersurfaces showsthat for general E as above, there is a reduced surface Σ E ⊆ T ,E whose specialization when E = E sp contains P ∪ P . Let E → C × B be a family of vector bundles on C parameterized by a smoothcurve B , with general fiber E and special fiber E sp . Denote by E b the restriction of E to C × { b } .Property ( ∗ ) then implies that φ (Σ E b ) for general b ∈ B is contained in at most one quadric Q E b in P . We argue by contradiction and assume that the quartic D , E b is a pull-back φ − ( Q ) for general b . One thus must have Q = Q E b . Next, the determinantal quartic D , E b is singular along T , E b , hencealong Σ E b . Let b (cid:55)→ q E b ∈ Sym I C (2), where q E b is a defining equation for the quadric Q E b . Then wefind that the first order derivative of the family φ ∗ q E b at b also vanishes along Σ E b , hence it must beproportional to φ ∗ q E b . We then conclude that the quadric Q E b is in fact constant, and thus must beequal to the quadric Q E sp . We now reach a contradiction by proving the following lemma. (cid:3) Lemma 26.
If the determinantal quartic D , E b is constant, equal to D sp = Q ∪ Q , then the vectorbundle E b on C does not deform with b ∈ B .Proof. Denoting V ,b := H ( C, E b ), V ,b := H ( C, E b ( η )), we have the multiplication map V ,b ⊗ V ,b → H ( C, K C ⊗ η )which is surjective for generic b since it is surjective for E = D ⊕ ( K C ⊗ D ∨ ) (see Lemma 23). Thedeterminantal quartic D , E b is the vanishing locus of the determinant of the corresponding bundlemap σ b : V ,b ⊗ O P ( H ( C,K C ( η )) ∨ ) → V ∨ ,b ⊗ O P ( H ( C,K C ( η )) ∨ ) (1) (5.29)on P (cid:0) H ( C, K C ⊗ η ) ∨ (cid:1) . We know that D , E b = Q ∪ Q for any b ∈ B , where the quadrics Q i aresingular (of rank 4), but with singular locus P i not intersecting C ⊆ Q ∩ Q . The morphism σ b hasrank exactly 1 generically along each Q i and the kernel of σ | D ,b determines a line bundle K i,b on itssmooth locus Q i \ P i . This line bundle is independent of b since Pic( Q i \ P i ) has no continuous part.The restriction of K i,b to C is thus constant. Finally, on the smooth part of ( Q ∩ Q ) reg , the kernelKer( σ ) contains the two line bundles K i,b | Q ∩ Q . Restricting to C ⊆ ( Q ∩ Q ) reg , we conclude thatKer σ b | C contains K i, | C for i = 0 ,
1. For b = 0, one hasKer σ | C = K , | C ⊕ K , | C and this thus remains true for general b . Finally, it follows from the construction and the fact that E b is generated by its sections that Ker σ b | C = E ∨ b , which finishes the proof. (cid:3)
6. Miscellany with anontrivial syzygy We now comment on an interesting rank 2 vector bundle appearing in our situation. Again, let φ L : C (cid:44) → P be a paracanonical curve of genus 8. We assume L is scheme-theoretically cut out byquadrics. Denoting by N C the normal bundle of C in the embedding in P , we consider the natural
6. Miscellany
6. Miscellany map I C (2) ⊗ O C → N ∨ C ⊗ L ⊗ (which is surjective by our assumption) given by differentiation along φ L ( C ), and let F denote its kernel. We thus have the short exact sequence:0 −→ F −→ I C (2) ⊗ O C −→ N ∨ C ⊗ L ⊗ −→ . (6.30)If K , ( C, L ) (cid:54) = 0, the map µ : I C (2) ⊗ H ( P , O (1)) → I C (3) is not surjective, hence not injective. Afortiori, the map µ : I C (2) ⊗ H ( P , O P (1)) → H ( C, N ∨ C ⊗ L ⊗ )induced by (6.30) is not injective, so that h ( C, F ( L )) (cid:54) = 0. In fact, the equivalence between thestatements h ( C, F ( L )) (cid:54) = 0 and K , ( C, L ) (cid:54) = 0 follows from the same argument once we know thatthere is no cubic polynomial on P vanishing with multiplicity 2 along C .We observe now that F is a vector bundle of rank 2 on the curve C , with determinant equal todet N C ⊗ L ⊗ ( − ∼ = K C ⊗ L ⊗ ( − . Hence if F ( L ) has a nonzero section, assuming this section vanishesnowhere along C , then F ( L ) is an extension of K C ⊗ L ∨ by O C . This provides an extension class e ∈ H ( C, L ⊗ K ∨ C ) = H ( C, K ⊗ C ⊗ L ∨ ) ∨ . (6.31)Assume now L ⊗ K ∨ C =: η is a nonzero 2-torsion element of Pic ( C ). Then e ∈ H ( C, L ) ∨ . On the other hand, according to Theorem 20, there exists a nontrivial syzygy γ = (cid:88) i =1 (cid:96) i ⊗ q i ∈ K , ( C, L ) = Ker (cid:8) H ( P , O P (1)) ⊗ I C (2) → I C (3) (cid:9) , which is degenerate by Proposition 13. As we saw already, it has in fact rank 6 for generic [ C, η ],hence determines a nonzero element f ∈ H ( P , O P (1)) ∨ = H ( C, L ) ∨ = H ( C, K C ⊗ L ∨ ) = H ( C, L ⊗ K ∨ C ) , (6.32)which is well-defined up to a coefficient. Proposition 27.
The two elements e and f are proportional.Proof. Equivalently, we show that the kernels of the two linear forms e, f ∈ H ( C, L ) ∨ are equal.Viewing γ as an element of Hom ( I C (2) ∨ , H ( C, L )), we have Ker( f ) = Im( γ ). On the other hand,the kernel of e identifies withIm (cid:110) j : H ( C, F ⊗ L ⊗ ⊗ K ∨ C ) → H ( C, L ) (cid:111) , where the map j is obtained by twisting the exact sequence 0 → O C → F ( L ) → K C ⊗ L ∨ → K C . We have F ⊗ L ⊗ ⊗ K ∨ C ∼ = F ∨ since det F ∼ = K C ⊗ L ⊗ ( − , hence there is a natural morphism i ∗ : I C (2) ∨ ⊗ O C → F ∨ ∼ = F ( L ⊗ ⊗ K ∨ C )dual to the inclusion F (cid:44) → I C (2) ⊗ O C of (6.30). The proposition follows from the following claim: Claim.
The morphism α : I C (2) ∨ → H ( C, L ) is equal to j ◦ i ∗ . Forgetting about the last identification F ∨ ∼ = F ⊗ L ⊗ ⊗ K ∨ C ), the claim amounts to the followinggeneral fact: For an evaluation exact sequence on a variety X −→ G −→ W ⊗ O X −→ M −→ s ∈ H ( X, G ( L )) = H ( X, H om ( G ∨ , L )) giving an element s (cid:48) ∈ Ker (cid:110) W ⊗ H ( X, L ) → H ( X, M ⊗ L ) (cid:111) ⊆ Hom ( W ∨ , H ( X, L )) , the induced map s : H ( X, G ∨ ) → H ( X, L ) composed with the map W ∨ → H ( X, G ∨ ) equals themap s (cid:48) : W ∨ → H ( X, L ). (cid:3) . Colombo, G. Farkas, A. Verra & C. Voisin, Syzygies of Prym and paracanonical curves of genus Using the exact sequence (6.30) in the general case of a genus 8 paracanonical curve [
C, L ] ∈ P , weobtain: Lemma 28.
A section s ∈ H ( C, F ( L )) ⊆ I C,L (2) ⊗ H ( C, L ) = Hom (cid:0) I C,L (2) ∨ , H ( C, L ) (cid:1) of rank6, determines an element e ∈ | L − K C | .Proof. The multiplication by s ∈ H ( F ( L )) ⊆ I C,L (2) ⊗ H ( C, L ) = H ( I C,L (2) ∨ ⊗ L ) determines thenatural maps F ∨ → L and g s : I C (2) ∨ ⊗ O C → L sitting in the following diagram:0 −→ K er ( g s ) −→ I C (2) ∨ ⊗ O C −→ L −→ ↓ ↓ ↓ −→ L − K C −→ F ∨ −→ L −→ , where I C (2) ∨ ⊗ O C → F ∨ is the dual of the natural inclusion of (6.30). Passing to global sections weget the inclusion H ( K er ( g s )) = Ker (cid:8) I C,L (2) ∨ → H ( C, L ) (cid:9) (cid:44) → H (2 L − K C ), which by hypothesisin 1-dimensional hence it defines an element e ∈ | L − K C | . (cid:3) Via the exact sequence (6.30) we can also show directly the following result that has been used inSection 3:
Lemma 29.
If there is a spin curve D = C ∪ E (cid:44) → P of genus 22 and degree 21 containing the genus8 paracanonical curve [ C, L ] as in Lemma 3 , then H C, ( F ( L )) (cid:54) = 0 . If there is no cubic polynomialon P vanishing with multiplicity along C , then K , ( C, L ) (cid:54) = 0 .Proof. Let e = C ∩ E and recall c ( F ) = − L + K C and O C ( e ) = 2 L − K C . Note that I D (2) ⊆ I C (2)is 6-dimensional. Tensor then the first vertical exact sequence of the following diagram by L and passto global sections. 0 0 0 ↓ ↓ ↓ −→ L ∨ −→ I D (2) ⊗ O C −→ I D / ( I D ∩ I C )(2) −→ ↓ ↓ ↓ −→ F −→ I C (2) ⊗ O C −→ N ∨ C (2) −→ ↓ ↓ ↓ −→ O C ( − e ) −→ O C −→ O C | e −→ ↓ ↓ ↓ . (cid:3) We return to the proof of Theorem 20 given in Section 5. Consider now a general paracanonical curve[
C, K C ⊗ η ] ∈ P . For a rank 2 vector bundle on C of degree 14, with noncanonical determinant, theequation h ( C, E ) ≥ (cid:15) ∈ Pic ( C ), the equation h ( C, E ⊗ (cid:15) ) ≥ E . Given C , there are 29 = 4 g − E , and 8 = g parameters for (cid:15) . It follows that we have at least a 5-dimensional family of pairs ( E, (cid:15) ),such that h ( C, E ) ≥ h ( C, E ⊗ (cid:15) ) ≥ . (6.33)Furthermore, the construction of Section 5 (together with Proposition 5) shows that for a generaltriple ( C, E, (cid:15) ) as above, one has K , ( C, L ) (cid:54) = 0, where L := det E ⊗ (cid:15) . Assuming the map ( E, (cid:15) ) (cid:55)→ L References References is generically finite on its image, we constructed in this way a five dimensional family of paracanonicalline bundles L ∈ Pic ( C ) with a nontrivial syzygy: K , ( C, L ) (cid:54) = 0. This family has the followingproperty: Lemma 30. If L = det E ⊗ (cid:15) , where E satisfies (6.33), the line bundle K ⊗ C ⊗ L ∨ satisfies the sameproperty. The family above, which has dimension at least five, is thus invariant under the involution L (cid:55)→ K ⊗ C ⊗ L ∨ on P , whose fixed locus is the Prym moduli space R .Proof. This follows from Serre duality, replacing E with E ∨ ⊗ K C and E ⊗ (cid:15) by E ∨ ⊗ (cid:15) ∨ ⊗ K C plusthe fact that det ( E ∨ ⊗ K C ) ⊗ (cid:15) ∨ ∼ = K ⊗ C ⊗ det E ∨ ⊗ (cid:15) ∨ . (cid:3) One can ask in general the following question:
Question 31.
Is the divisor
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