Kirchhoff’s theorem for Prym varieties

Abstract

We prove an analogue of Kirchhoff’s matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel–Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition and prove that its global degree is $2^{g-1}$ . Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel–Prym map is $2^{g-1}$ as well.

Abelian tropical covers

“…Applications to algebraic Prym-Brill-Noether theory are studied in [LU21] and [CLRW22]. See [Len22] for a survey on tropical Prym varieties. In a similar vein, Song [Son19] considers G-invariant linear systems with the goal of studying their descent properties to the quotient.…”

“…In a different direction, Jensen and Len [ JL18 ] consider -covers of arbitrary tropical curves, and define the tropical Prym variety associated to such a cover. This object is equipped with a canonical polyhedral decomposition, leading to a combinatorial formula for its volume [ GZ23, LZ22 ]. A tropical version of Donagi’s n -gonal construction is investigated in [ RZ22 ].…”

Critical groups of strongly regular graphs and their generalizations

“…Zaslavsky formulated a Matrix-Tree Theorem for signed graphs that provides a combinatorial meaning of |K(G σ )| [28]; the tropical interpretation of these groups was studied by Len and Zakharov [19]. There are fewer works on the structure of critical groups of signed groups in the literature compared with ordinary graphs.…”

A Note on the Critical Groups of Strongly Regular Graphs and Their Generalizations

“…Zaslavsky formulated a Matrix-Tree Theorem for signed graphs that provides a combinatorial meaning of |K(G σ )| [27]; the tropical interpretation of these groups was studied by Len and Zakharov [18]. There are fewer works on the structure of critical groups of signed groups in the literature compared with ordinary graphs.…”