Integrable discrete Schrodinger equations and a characterization of Prym varieties by a pair of quadrisecants

TL;DR

This paper characterizes Prym varieties through geometric and theta-functional properties, and introduces a difference-differential analog of the Novikov-Veselov hierarchy related to integrable discrete Schrödinger equations.

Contribution

It provides a new geometric characterization of Prym varieties using quadrisecants and develops a novel difference-differential hierarchy for integrable discrete Schrödinger equations.

Findings

Prym varieties are characterized by symmetric pairs of quadrisecant planes.

Prym varieties are characterized by new theta-functional equations.

A difference-differential analog of the Novikov-Veselov hierarchy is constructed.

Abstract

We prove that Prym varieties are characterized geometrically by the existence of a symmetric pair of quadrisecant planes of the associated Kummer variety. We also show that Prym varieties are characterized by certain (new) theta-functional equations. For this purpose we construct and study a difference-differential analog of the Novikov-Veselov hierarchy.