The Moduli Space of Rank-3 Vector Bundles with Trivial Determinant over a Curve of Genus 2 and Duality

Let SUX(3) be the moduli space of semi-stable vector bundles of rank 3 and trivial determinant on a curve X of genus 2. It maps onto P and the map is a double cover branched over a sextic hypersurface called the Coble sextic. In the dual P there is a unique cubic hypersurface, the Coble cubic, singular exactly along the abelian surface of degree 1 line bundles on X. We give a new proof that these two hypersurfaces are dual. As an immediate corollary, we derive a Torelli-type result. Introduction Let us fix once and for all a smooth projective curve X of genus g = 2. We denote by J(X), or even by J since we fixed X, the variety parametrizing classes of line bundles (or divisors) on X of degree d. When d = 0, we write J for the Jacobian of X. The variety J1 carries a canonical Riemann theta divisor Θ = {L ∈ J : H(X,L) 6= 0}. Moreover, we know that 3Θ is very ample on J1, so this give an embedding of J1 into P8 = |3Θ|. A. Coble in [Cob17] shows that J1 is set-theoretically cut out by 9 quadrics. In [Bar95], W. Barth proves that this is even a scheme-theoretic intersection. In particular, if we denote by IJ1 the ideal sheaf of J 1 in P8, then (0.1) dimH(P,IJ1(2)) = 9, which can also be derived from the projective normality of the embedding of J1 [Koi76]. It turns out that the quadrics are the partial derivatives of a cubic, so there is a unique cubic hypersurface C3 singular exactly along the Abelian surface J1. The hypersurface C3 has hence been dubbed the Coble cubic. Let SUX(3) be the moduli space of semi-stable vector bundles of rank 3 and trivial determinant on a curve X of genus 2. It maps onto P8 = |3Θ| and the map is a double cover branched over a sextic hypersurface C6. This P8 is the dual P8 of the one in which C3 lies. I. Dolgachev conjectured that C3 and C6 are dual varieties, and by analogy with the case of the Coble quartic, C6 is known as the Coble sextic. Indeed, the Coble quartic, a quartic hypersurface in P7, has an interpretation as the moduli space SUC(2) of semi-stable vector bundles of 2 and trivial determinant on a fixed non-hyperelliptic curve C of genus 3 (see [NR87]). Moreover, this quartic hypersurface is singular exactly along the Kummer surface associated to C (and 2000 Mathematics Subject Classification. Primary 14H60, 14E05, 14J70. 1 2 NGUY ̃̂ EN QUANG MINH embedded into P7) and thanks to its moduli space interpretation, C. Pauly [Pau02] proved that the Coble quartic is self-dual. In this paper, we prove the following theorem: Theorem 5.1. The Coble hypersurfaces C3 and C6 are dual. The result was first proved by A. Ortega Ortega [Ort03] in her thesis. We give here a different proof, which uses a more thorough study of the variety C6 and a more general description of the dual map in terms of the vector bundles. In particular we compute the degree of its singular locus. As a corollary, we derive a non-abelian Torelli result. Acknowledgements I would like to express my sincere thanks to Igor Dolgachev for all the support, guidance and encouragement during the research leading to this paper. I would also like to acknowledge the insightful discussions with Alessandro Verra, Angela Ortega Ortega, Mihnea Popa and Ravi Vakil. 1. Definitions and preliminaries For a vector bundle E or rank n on X, we define its determinant