Determinantal characterization of canonical curves and combinatorial theta identities

We characterize genus g canonical curves by the vanishing of combinatorial products of g + 1 determinants of Brill-Noether matrices. This also implies the characterization of canonical curves in terms of (g−2)(g−3)/2 theta identities. A remarkable mechanism, based on a basis of H0(KC) expressed in terms of Szegö kernels, reduces such identities to a simple rank condition for matrices whose entries are logarithmic derivatives of theta functions. Such a basis, together with the Fay trisecant identity, also leads to the solution of the question of expressing the determinant of Brill-Noether matrices in terms of theta functions, without using the problematic Klein-Fay section σ.