×

zbMATH — the first resource for mathematics

Torelli theorem for graphs and tropical curves. (English) Zbl 1200.14025
The Torelli map associates the Jacobian to an algebraic curve. Via dual graphs respectively tropical curves, algebraic curves are connected to graphs. A graph analogue of the Jacobian of a curve, namely the Jacobian torus respectively the dual Albanese torus, has been defined by M. Kotani and T. Sunada [Adv. Appl. Math. 24, No. 2, 89–110 (2000; Zbl 1017.05038)]. This paper is concerned with a Torelli theorem for graphs and tropical curves, i.e. it answers the question when two graphs are such that their Albanese tori are isomorphic. Assuming some facts about the moduli space of genus \(g\) tropical curves and the moduli space of tropical Jacobians, the authors also prove that the tropical Torelli map is of degree one onto its image. The assumptions made here have meanwhile been settled in a preprint of Branetti, Melo and Viviani.

MSC:
14C34 Torelli problem
14T05 Tropical geometry (MSC2010)
05C99 Graph theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] V. Alexeev, Compactified Jacobians and Torelli map , Publ. Res. Inst. Math. Sci. 40 (2004), 1241-1265. · Zbl 1079.14019
[2] I. V. Artamkin, The discrete Torelli theorem (in Russian), Mat. Sb. 197 (2006), 3-16.; English translation in Sb. Math. 197 (2006), 1109-1120. · Zbl 1184.14038
[3] R. Bacher, P. De La Harpe, and T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph , Bull. Soc. Math. France 125 (1997), 167-198. · Zbl 0891.05062
[4] M. Baker and S. Norine, Riemann-Roch and Abel-Jacobi theory on a finite graph , Adv. Math. 215 (2007), 766-788. · Zbl 1124.05049
[5] N. L. Biggs, Algebraic potential theory on graphs , Bull. London Math. Soc. 29 (1997), 641-682. · Zbl 0892.05033
[6] L. Caporaso, Geometry of the theta divisor of a compactified Jacobian , J. Eur. Math. Soc. (JEMS) 11 (2009), 1385-1427. · Zbl 1202.14030
[7] L. Caporaso and F. Viviani, Torelli theorem for stable curves , · Zbl 1230.14037
[8] R. Diestel, Graph Theory , Grad. Texts in Math. 173 , Springer, New York, 1997.
[9] A. Gathmann, Tropical algebraic geometry , Jahresber. Dtsch. Math.-Verein. 108 (2006), 3-32. · Zbl 1109.14038
[10] M. Kotani and T. Sunada, Jacobian tori associated with a finite graph and its abelian covering graphs , Adv. in Appl. Math. 24 (2000), 89-110. · Zbl 1017.05038
[11] G. Mikhalkin, Enumerative tropical algebraic geometry in \(\R^2\) , J. Amer. Math. Soc. 18 (2005), 313-377. · Zbl 1092.14068
[12] -, “Tropical geometry and its applications” in International Congress of Mathematicians, Vol. II , Eur. Math. Soc., Zürich, 2006, 827-852. · Zbl 1103.14034
[13] -, What is\(\dots\)a tropical curve? Notices Amer. Math. Soc. 54 (2007), 511-513. · Zbl 1142.14300
[14] -, Introduction to tropical geometry: Notes from the IMPA lectures, summer 2007 ,
[15] G. Mikhalkin and I. Zharkov, “Tropical curves, their Jacobians and theta functions” in Curves and Abelian Varieties , Contemp. Math. 465 , Amer. Math. Soc., Providence, 2008, 203-231. · Zbl 1152.14028
[16] Y. Namikawa, A new compactification of the Siegel space and degeneration of Abelian varieties, II , Math. Ann. 221 (1976), 201-241. · Zbl 0327.14013
[17] T. Oda and C. S. Seshadri, Compactifications of the generalized Jacobian variety , Trans. Amer. Math. Soc. 253 (1979), 1-90. · Zbl 0418.14019
[18] J. G. Oxley, Matroid Theory , Oxford Sci. Publ., Oxford Univ. Press, New York, 1992.
[19] J. Richter-Gebert, B. Sturmfels, and T. Theobald, “First steps in tropical geometry” in Idempotent Mathematics and Mathematical Physics (Vienna, 2003) , Contemp. Math. 377 , Amer. Math. Soc., Providence, 2005. · Zbl 1093.14080
[20] H. E. Robbins, Questions, discussions, and notes: A theorem on graphs with an application to a problem of traffic control , Amer. Math. Monthly 46 (1939), 281-283. · Zbl 0021.35703
[21] H. Whitney, \(2\)-isomorphic graphs , Amer. J. Math. 55 (1933), 245-254. JSTOR: · Zbl 0006.37005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.