A geometric characterization of Prym varieties -- Samuel Grushevsky

Prym varieties are a special class of abelian varieties, naturally embedded in Jacobians of curves having an involution. Many of geometric properties of Jacobians can be generalized to Prym varieties. In analogy with the case of Welters' trisecant conjecture - that Jacobians are characterized by their Kummer variety having a trisecant line - recently proven by Krichever, we prove that Pryms are characterized by their Kummer variety having a "symmetric" pair of quadrisecant planes. We will explain the ideas of the integrable-systems approach leading to proving this, and the underlying geometry of the problem. Based on joint work with Igor Krichever.