Kirchhoff's theorem for Prym varieties

2020/12/31 by Yoav Len, Dmitry Zakharov
Computer Science · Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic geometry #Algebraic number #Algebraic variety #Commutative Algebra and Its Applications #Degree (music) #Mathematical analysis #Mathematics #Polynomial and algebraic computation #Pure mathematics #Variety (cybernetics) #math.AG #math.CO #msc:14H40 #msc:14T15 #msc:14T20

paper · pdf · doi:10.1017/fms.2021.75

published as Forum of Mathematics, Sigma (2022), Vol. 10:e11 1-54 · 58 pages, 13 figures, Appendix by Sebastian Casalaina-Martin. Added a detailed example of the harmonic structure of the Abel-Prym map

openalex publication_date 2022/01/01 · arxiv created 2022/03/06 · arxiv updated 2022/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• Geometry of Algebraic Curves
• Moduli of representations of the fundamental group of a smooth projective variety II
• The Intermediate Jacobian of the Cubic Threefold
• A universal tropical Jacobian over M_g^\trop
• Moduli of representations of the fundamental group of a smooth projective variety I