Curves and symmetric spaces

Abstract

Let Sym3 C !P .k Sym3 k Sym3 k k/DP; A 7! .1 WA WA WdetA/ be the Veronese embedding of the space of symmetric matrices of degree 3, where A is the cofactor matrix of A. The closure SpG.3; 6/ of this image is a 6-dimensional homogeneous variety of the symplectic group Sp.3/. A canonical curve C16 P of genus 9 over a perfect field k is isomorphic to a complete linear section of this projective variety SpG.3; 6/ P unless C k N k, N k being the algebraic closure, is a covering of degree at most 5 of the projective line. We prove this by means of linear systems of higher rank.