Curves and Symmetric Spaces, II

We describe the canonical model of an algebraic curve of genus 9 over a perfect field when the Clifford index is maximal (=3) by means of linear systems of higher rank. Let SpG(n, 2n) be the symplectic Grassmannian, that is, the Grassmannian of Lagrangian subspaces of a 2n-dimensional symplectic vector space, over a field k. In the case n = 3, SpG(3, 6) is a 6-dimensional homogeneous variety and (equivariantly) embedded into the projective space P with homogeneous coordinate (y : X : Y : x), where x, y ∈ k are scalars and X, Y ∈ Sym3 k are symmetric matrices. Then SpG(3, 6) ⊂ P is the common zero locus of the 21 (=6+6+9) quadratic equations