Curves and symmetric spaces, II

Abstract

Let Sym 3 C →P * (k⊕Sym 3 k⊕Sym 3 k⊕k) = P 13 , A ↦ (1:A:A':det A) 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 C 16 C P 8 of genus 9 over a perfect field k is isomorphic to a complete linear section of this projective variety SpG(3, 6) C P 13 unless C ⊗ k k, 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.