Curves and symmetric spaces, II

Abstract

Let ${\rm Sym}_3\, \mathbf{C} \longrightarrow \mathbf{P}_*(k \oplus {\rm Sym}_3\,k \oplus {\rm Sym}_3\,k \oplus k) = {\bf P}^{13}, A \mapsto (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 $\operatorname{SpG}(3, 6)$ of this image is a 6-dimensional homogeneous variety of the symplectic group $\operatorname{Sp}(3)$. A canonical curve $C_{16} \subset {\bf P}^8$ of genus 9 over a perfect field $k$ is isomorphic to a complete linear section of this projective variety $\operatorname{SpG}(3, 6) \subset {\bf P}^{13}$ unless $C \otimes_k \bar k$, $\bar k$ 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.