TGiZ-Seminar: Tropical geometry in Zoom (Third meeting Summer Semester 2022)
Dmitry Zakharov (Central Michigan University)

Abstract: The Jacobian of a finite graph is a finite abelian group, and Kirchhoff’s celebrated matrix tree theorem computes the order of the Jacobian as the number of spanning trees of the graph. The Jacobian Jac(G) of a metric graph G is a real torus of dimension equal to b_1(G), and a weighted version of Kirchhoff’s theorem expresses the volume of Jac(G) as a weighted sum over all spanning trees of G.

A recent paper of An, Baker, Kuperberg, and Shokrieh gives a geometric interpretation of the weighted matrix-tree theorem of a metric graph G, based on an earlier result of Mikhalkin and Zharkov. Namely, each element of Jac(G) is represented by a unique (up to translation) so-called break divisor. The type of break divisor defines a canonical cellular decomposition of Jac(G), and the individual terms in the volume formula for Jac(G) are the volumes of the cells.

I will state and prove analogous results for the tropical Prym variety Pr(G’/G) associated to a double cover of metric graphs G’->G, as defined by Jensen, Len, and Ulirsch. The volume of Pr(G’/G) is calculated as a weighted sum over certain collections of spanning cycles on the target graph G, generalizing a similar result of Zaslavsky, Reiner and Tseng for ordinary graphs. I will then give a geometric interpretation of the volume formula in terms of a semi-canonical representability result for Prym divisors. I will discuss possible applications to the problem of resolving the Prym-Torelli map.