Kirchhoff’s theorem for Prym varieties

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 pa...

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Veröffentlicht in:Forum of Mathematics, Sigma Jg. 10
Hauptverfasser: Len, Yoav, Zakharov, Dmitry
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cambridge, UK Cambridge University Press 01.01.2022
Schlagworte:
Algebra
Algebraic and Complex Geometry
Decomposition
Graphs
Theorems
Trees
14T05
14H40
ISSN:2050-5094, 2050-5094
Online-Zugang:Volltext
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Zusammenfassung: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.
Bibliographie:ObjectType-Article-1
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ObjectType-Feature-2
content type line 14
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2021.75