  We give 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.
Here we treat the case of flexes and degenerate trisecants. The basic tool we
use is a theorem we prove asserting that the base locus of the linear system
associated to an effective line bundle on an abelian variety is reduced. This
result allows us to remove all the extra assumptions that were introduced in
the theorems by the first author, C. De Concini, G.Marini, and O. Debarre, in
order to achieve algebro-geometric proofs of the results above.
