On the even-order vanishing of Jacobian theta functions

Characterization of Jacobian varieties in terms of soliton equations

“…The r-functions satisfy, and are characterized by, infinitely many Hirota bilinear equations, which are given by the following generating function expansion [3, formula (1.7.1)]: (8) shows various geometric properties of the theta divisor of a Jacobian (see, e.g., [9]). Hirota's bilinear operator intertwines the diagonal action of G t on the space of pairs of functions and the action of G~ = {p2; peG1 } =G l on the space of functions :…”

A Special Case of the Γ00 Conjecture

“…In fact a slightly more precise result is known for Jacobians: from the explicit formulas for Jacobians in [7] and [11] more information can be obtained than just the statement that C − C ⊂ Bs(Γ 00 ) (see proposition 1 for the precise formulation). In this paper we prove that this refined version of the Γ 00 conjecture (a certain extra condition on the linear dependence, see theorem 1 for a precise formulation) characterizes Jacobians.…”

Abelian solutions of the soliton equations and Riemann-Schottky problems

“…Prym varieties possess generalizations of some properties of Jacobians. In [7] Beauville and Debarre, and in [22] Fay showed that the Kummer images of Prym varieties admit a 4-dimensional family of quadrisecant planes (as opposed to a 4dimensional family of trisecant lines for Jacobians). Similarly to the case of Jacobians, it was then shown by Debarre in [12] that the existence of a one-dimensional family of quadrisecants characterizes Prym varieties among all ppavs.…”