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Self duality of Coble's quartic hypersurface and applications

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22 pages
Self-duality of Coble's quartic hypersurface and applications Christian Pauly January 11, 2008 Abstract The moduli space M0 of semi-stable rank 2 vector bundles with fixed trivial determinant over a non-hyperelliptic curve C of genus 3 is isomorphic to a quartic hypersurface in P7 (Coble's quartic). We show that M0 is self-dual and that its polar map associates to a stable bundle E ? M0 a bundle F which is characterized by dimH0(C,E?F ) = 4. The projective space PH0(C,E ? F ) is equipped with a net of quadrics ? and it is shown that the map which associates to E ? M0 the isomorphism class of the plane quartic Hessian curve of ? is a dominant map to the moduli space of genus 3 curves. 1 Introduction In his book [C] A.B. Coble constructs for any non-hyperelliptic curve C of genus 3 a quartic hypersurface in P7 which is singular along the Kummer variety K0 ? P7 of C. It is shown in [NR] that this hypersurface is isomorphic to the moduli space M0 of semi-stable rank 2 vector bundles with fixed trivial determinant. For many reasons Coble's quartic hypersurface may be viewed as a genus-3-analogue of a Kummer surface,i.e. a quartic surface S ? P3 with 16 nodes.

  • theta-divisors

  • classifying map

  • m˜0 ? ?

  • k0 ?

  • let m0

  • ??

  • then ∆

  • stable bundle

  • space tem0


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Self-duality
1
of
Coble’s quartic hypersurface
Christian Pauly
January 11, 2008
Abstract
and
applications
The moduli spaceM0of semi-stable rank 2 vector bundles with fixed trivial determinant over a non-hyperelliptic curveCof genus 3 is isomorphic to a quartic hypersurface inP7 (Coble’s quartic). We show thatM0is self-dual and that its polar map associates to a stable bundleE∈ M0a bundleFwhich is characterized by dimH0(C, EF The projective) = 4. spacePH0(C, EFΠ and it is shown that the map) is equipped with a net of quadrics which associates toE∈ M0the isomorphism class of the plane quartic Hessian curve of Π is a dominant map to the moduli space of genus 3 curves.
Introduction
In his book [C] A.B. Coble constructs for any non-hyperelliptic curveCof genus 3 a quartic hypersurface inP7which is singular along the Kummer varietyK0P7ofC. It is shown in [NR] that this hypersurface is isomorphic to the moduli spaceM0of semi-stable rank 2 vector bundles with fixed trivial determinant. For many reasons Coble’s quartic hypersurface may be viewed as a genus-3-analogue of a Kummer surface,i.e. a quartic surfaceSP3with 16 nodes. example For the restriction ofM0to an eigenspaceP3αP7for the action of a 2-torsion pointαJ C[2] is isomorphic to a Kummer surface (of the corresponding Prym variety). It is classically known (see e.g. [GH]) that a Kummer surfaceSP3is self-dual.
In this paper we show that this property also holds for the Coble quarticM0(Theorem 3.1). The rational polar mapD:P7−→(P7)mapsM0birationally toMω(P7), whereMω (=M0 More) is the moduli space parametrizing vector bundles with fixed canonical determinant. precisely we show that the embedded tangent space at a stable bundleEtoM0corresponds to a semi-stable bundleD(E) =F∈ Mω, which is characterized by the condition dimH0(C EF) = 4 e (its maximum). We also show thatDresolves to a morphismDby two successive blowing-ups, and thatDcontracts the trisecant scroll ofK0to the Kummer varietyKω⊂ Mω. The condition which relatesEto its “tangent space bundle”F, namely dimH0(C EF) = 4, leads to many geometric properties. First we observe thatPH0(C EF) is naturally equipped with a net of quadrics Π whose base points (Cayley octad) correspond bijectively to the 8 line subbundles of maximal degree ofE(and ofF). The Hessian curve Hess(E) of the net of quadrics Π =|ω|is a plane quartic curve, which is everywhere tangent (Proposition 4.7) to the canonical curveC⊂ |ω|,i.e. Hess(E)C= 2Δ(E) for some divisor Δ(E)∈ |ω2|. Since these constructions invarian we introduce the quotientN=M0/J C[2] parametrizingPSL2-bundles over aCanreJdCemapatth3)th4n1.tioiposo(wrPwhoes,t]-[2N −Δ→ |ω2|,E7Δ(E) is the restriction of the projection from the projective space |L|N ⊂=P13(Lis the ample generator of Pic(N)) with
1
center of projection given by the linear span of the Kummer varietyK0⊂ N(K0parametrizes decomposablePSL2-bundles). We also show (Corollary 4.16) that the Hessian map RN →,E7→Hess(E) is finite of degree 72, whereRis the rational space parametrizing plane quartics everywhere tangent toC⊂ |ω|= P2 the isomorphism class of Hess(. ConsideringE), we deduce that the map Hess : MN →3 is dominant, whereM3 actually prove thatis the moduli space of smooth genus 3 curves. We e some Galois-covers NN →andPC→ R In particular weare birational (Proposition 4.15). e endow the spaceN, parametrizingPSL2-bundlesEwith an ordered set of 8 line subbundles ofE of maximal degree, with an action of the Weyl groupW(E7) such that the action of the central elementw0W(E7) coincides with the polar mapD. We hope that these results will be useful for dealing with several open problems, e.g. rationality of the moduli spacesM0andN. I would like to thank S. Ramanan for some inspiring discussions on Coble’s quartic.
2 The geometry of Coble’s quartic
In this section we briefly recall some known results related to Coble’s quartic hypersurface, which can be found in the literature, e.g. [DO], [L2], [NR], [OPP]. We refer to [B1], [B2] for the results on the geometry of the moduli of rank 2 vector bundles.
2.1 Coble’s quartic as moduli of vector bundles LetCbe a smooth non-hyperelliptic curve of genus 3 with canonical line bundleω Pic. Letd(C) be the Picard variety parametrizing degreedline bundles overCandJ C:= Pic0(C) be the Jacobian variety. We denote byK0the Kummer variety ofJ Cand byKωthe quotient of Pic2(C) by the involutionξ7→ωξ1 Θ. LetPic2(C) be the Riemann Theta divisor and let Θ0J Cbe a symmetric Theta divisor, i.e. a translate of Θ by a theta-characteristic. We also recall that the two linear systems||and|0|are canonically dual to each other via Wirtinger duality ([Mu2] p. 335),i.e. we have an isomorphism||=|0|. LetM0(resp.Mωof semi-stable rank 2 vector bundles over) denote the moduli space Cwith fixed trivial (resp. canonical) determinant. The singular locus ofM0is isomorphic toK0and points inK0correspond to bundlesEwhoseS-equivalence class [E] contains a decomposable bundle of the formMM1forMJ C. We have natural morphisms M0D→ ||=P7MωD→ |0|=||which send a stable bundleE∈ M0to the divisorD(E) whose support equals the set{LPic2(C)|dimH0(C EL)>0}(ifE∈ Mω, replace Pic2(C) byJ C the semi-stable). On boundaryK0(resp.Kω) the morphismD The moduli spacesrestricts to the Kummer map. M0andMωare isomorphic, although non-canonically (consider tensor product with a theta-characteristic). It is known that the Picard group Pic(M0) isZand that|L|=||, whereLis the ample generator of Pic(M0). The main theorem of [NR] asserts thatDembedsM0as a quartic hypersurface in||=P7, which was originally described by A.B. Coble [C] (section 33(6)). Coble’s quartic is characterized
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