**Date/Time**

Date(s) - 17/03/2023*3:30 pm - 4:30 pm*

**Date & Time: **March 17, 2023 at 3:30-4:30 pm

**Speaker:** Dmitry Zakharov, Central Michigan University

**Title:** *The tropical Prym variety*

**Abstract:** Tropical geometry studies discrete, piecewise-linear analogues of algebraic objects. This correspondence is particularly well-behaved for two classes of objects. The tropical analogue ofan algebraic curve is a metric graph, while the analogue of a complex abelian variety is a real torus with additional integral structure. There are two standard ways to assign abelian varieties to algebraic curves. The Jacobian variety of a curve of genus g is an abelian variety of dimension g. Given an unramified double cover of a genus g curve, the kernel of the norm map on the Jacobians is an abelian variety of dimensiong-1, called the Prym variety of the double cover. In my talk, I will talk about the tropical version of the Prym construction, which associates a tropical abelian variety to a double cover of metric graphs. I will prove a volume formula for the tropical Prym variety, which is an analogue of Kirchhoff’s matrix tree theorem. This formula has a geometric interpretation in terms of a canonical polyhedral decomposition of the tropical Prym variety (a similar decomposition for the tropical Jacobian was found by Mikhalkin—Zharkov and An—Baker—Kuperberg—Shokrieh). I will also discuss the tropical analogues of the trigonal and tetragonal constructions of Recillas and Donagi.

Joint work with Yoav Len and Felix Roehrle.

**Location: **Hamilton Hall 305

*Refreshments available at 3:00 pm in Hamilton Hall Lounge*