Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebro-geometric approach

In this talk, I will present a completely algebro-geometric proofs of a theorem by T. Shiota, and of a theorem by I. Krichever, characterizing Jacobians of algebraic curves among all irreducible principally polarized abelian varieties. Shiota's characterization is given in terms of the KP equation. Krichever's characterization is given in terms of trisecant lines to the Kummer variety. I will treat the case of flexes and degenerate trisecants. The basic tool that I will use is a theorem asserting that the base locus of the linear system associated to an effective line bundle on an abelian variety is reduced. This result will allow me to remove all the extra assumptions that were introduced in the theorems by E. Arbarello, C. De Concini, G.Marini, and O. Debarre, in order to achieve algebro-geometric proofs of the results above. This is a joint work with E. Arbarello and G. Pareschi.