Algebraic curves have a discrete analog in finite graphs. Pursuing this analogy, we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for 3-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. By contrast, we prove that, in a suitably defined sense, the tropical Torelli map has degree one. Finally, we describe some natural posets associated to a graph and prove that they characterize its Delaunay decomposition.

Caporaso, L., Viviani, F. (2010). Torelli theorem for graphs and tropical curves. DUKE MATHEMATICAL JOURNAL, 153(1), 129-171 [10.1215/00127094-2010-022].

Torelli theorem for graphs and tropical curves

CAPORASO, Lucia;VIVIANI, FILIPPO
2010-01-01

Abstract

Algebraic curves have a discrete analog in finite graphs. Pursuing this analogy, we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for 3-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. By contrast, we prove that, in a suitably defined sense, the tropical Torelli map has degree one. Finally, we describe some natural posets associated to a graph and prove that they characterize its Delaunay decomposition.
2010
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/03
English
Con Impact Factor ISI
Caporaso, L., Viviani, F. (2010). Torelli theorem for graphs and tropical curves. DUKE MATHEMATICAL JOURNAL, 153(1), 129-171 [10.1215/00127094-2010-022].
Caporaso, L; Viviani, F
Articolo su rivista
File in questo prodotto:
File Dimensione Formato  
TorelliGraphDMJprint.pdf

solo utenti autorizzati

Tipologia: Versione Editoriale (PDF)
Licenza: Copyright dell'editore
Dimensione 364.37 kB
Formato Adobe PDF
364.37 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/360845
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 36
  • ???jsp.display-item.citation.isi??? 35
social impact