Algebraic Geometry Seminar - Dmitry Zakharov

Title: Kirchhoff’s theorem and semi-canonical representatives for the tropical Prym variety

 

Speaker: Dmitry Zakharov - Central Michigan University

 

Abstract: The Jacobian of a finite graph $G$ is a finite group, defined as the group of degree zero divisors on the vertices of $G$ modulo linear equivalence. Kirchhoff’s celebrated matrix tree theorem states that the order of the Jacobian of $G$ is equal to the number of spanning trees. The Jacobian a metric graph $\Gamma$, defined similarly, is a real torus of dimension equal to the first Betti number of $\Gamma$, and a weighted version of Kirchhoff’s theorem expresses the volume of the Jacobian of $\Gamma$ as a weighted sum over all spanning trees of $\Gamma$.

 

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

 

 I will state and prove analogous results for the tropical Prym variety associated to a double cover of metric graphs $p:\Gamma’\to \Gamma$, as defined by Jensen, Len, and Ulirsch. The volume of the Prym variety is calculated as a weighted sum over certain spanning cycles on the target graph $\Gamma$. The volume formula has a geometric interpretation in terms of semi-canonical representations for Prym divisors.

 

 Joint work with Yoav Len.