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.

arXiv comment58 pages, 13 figures, Appendix by Sebastian Casalaina-Martin. Added a detailed example of the harmonic structure of the Abel-Prym map

References in corpus (13)

• Chip-firing games, potential theory on graphs, and spanning trees
• Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem
• Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory
• Tropicalization of theta characteristics, double covers, and Prym varieties
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• Non-Archimedean and tropical theta functions
• Brill-Noether theory for cyclic covers
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• Enumerative geometry of elliptic curves on toric surfaces
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• A universal tropical Jacobian over $M_g^{\trop}$

Cited by in corpus (1)

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