Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities

Details Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities

2006-05-30

arxiv.org/abs/math/0605734v3

Mathematics

Algebraic Geometry

35 pages. New results, presentation improved, clarifications added. Accepted for publication in Math. Ann

Scientific paper

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 H^0(K_C) expressed in terms of Szego 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 sigma.

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