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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| Published in: | Forum of Mathematics, Sigma Vol. 10 |
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| Main Authors: | , |
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| Language: | English |
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Cambridge University Press
01.01.2022
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| ISSN: | 2050-5094, 2050-5094 |
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| 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. |
|---|---|
| AbstractList | 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. 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. |
| ArticleNumber | e11 |
| Author | Zakharov, Dmitry Len, Yoav |
| Author_xml | – sequence: 1 givenname: Yoav orcidid: 0000-0002-4997-6659 surname: Len fullname: Len, Yoav email: yoav.len@st-andrews.ac.uk organization: 1Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, UK; E-mail: yoav.len@st-andrews.ac.uk – sequence: 2 givenname: Dmitry surname: Zakharov fullname: Zakharov, Dmitry email: dvzakharov@gmail.com organization: 2Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA; E-mail: dvzakharov@gmail.com |
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| Snippet | 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... |
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| SubjectTerms | Algebra Algebraic and Complex Geometry Decomposition Graphs Theorems Trees |
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| Title | Kirchhoff’s theorem for Prym varieties |
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