A Torelli theorem for graphs via quasistable divisors
The Torelli theorem establishes that the Jacobian of a smooth projective curve, together with the polarization provided by the theta divisor, fully characterizes the curve. In the case of nodal curves, there exists a concept known as fine compactified Jacobian. The fine compactified Jacobian of a cu...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Online Access: | https://www.cambridge.org/core/product/identifier/S205050942400135X/type/journal_article |
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| author | Alex Abreu Marco Pacini |
| author_facet | Alex Abreu Marco Pacini |
| author_sort | Alex Abreu |
| collection | DOAJ |
| description | The Torelli theorem establishes that the Jacobian of a smooth projective curve, together with the polarization provided by the theta divisor, fully characterizes the curve. In the case of nodal curves, there exists a concept known as fine compactified Jacobian. The fine compactified Jacobian of a curve comes with a natural stratification that can be regarded as a poset. Furthermore, this poset is entirely determined by the dual graph of the curve and is referred to as the poset of quasistable divisors on the graph. We present a combinatorial version of the Torelli theorem, which demonstrates that the poset of quasistable divisors of a graph completely determines the biconnected components of the graph (up to contracting separating edges). Moreover, we achieve a natural extension of this theorem to tropical curves. |
| format | Article |
| id | doaj-art-055690b473f64188a11c7ffb08cc37c1 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-055690b473f64188a11c7ffb08cc37c12025-02-03T10:39:38ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.135A Torelli theorem for graphs via quasistable divisorsAlex Abreu0Marco Pacini1Universidade Federal Fluminense, Instituto de Matemática e Estatística, Rua Prof. M. W. de Freitas, S/N, 24210-201 Niterói, Brazil;Universidade Federal Fluminense, Instituto de Matemática e Estatística, Rua Prof. M. W. de Freitas, S/N, 24210-201 Niterói, Brazil; E-mail:The Torelli theorem establishes that the Jacobian of a smooth projective curve, together with the polarization provided by the theta divisor, fully characterizes the curve. In the case of nodal curves, there exists a concept known as fine compactified Jacobian. The fine compactified Jacobian of a curve comes with a natural stratification that can be regarded as a poset. Furthermore, this poset is entirely determined by the dual graph of the curve and is referred to as the poset of quasistable divisors on the graph. We present a combinatorial version of the Torelli theorem, which demonstrates that the poset of quasistable divisors of a graph completely determines the biconnected components of the graph (up to contracting separating edges). Moreover, we achieve a natural extension of this theorem to tropical curves.https://www.cambridge.org/core/product/identifier/S205050942400135X/type/journal_article |
| spellingShingle | Alex Abreu Marco Pacini A Torelli theorem for graphs via quasistable divisors Forum of Mathematics, Sigma |
| title | A Torelli theorem for graphs via quasistable divisors |
| title_full | A Torelli theorem for graphs via quasistable divisors |
| title_fullStr | A Torelli theorem for graphs via quasistable divisors |
| title_full_unstemmed | A Torelli theorem for graphs via quasistable divisors |
| title_short | A Torelli theorem for graphs via quasistable divisors |
| title_sort | torelli theorem for graphs via quasistable divisors |
| url | https://www.cambridge.org/core/product/identifier/S205050942400135X/type/journal_article |
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