On the tropical Torelli map

By Silvia Brannetti, Margarida Melo, Filippo Viviani. 2009

130 citations recorded by the source corpus.

Abstract We construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9af35f4f2810396120411f01a6db4c994051e929

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Citations in the corpus

Algebraic and tropical curves: comparing their moduli spaces (2011, methodology): “We constructed the bordification of the moduli space of pure pointed tropical curves using weighted tropical curves (as done in [5] for the unpointed case, using a different set up) and its compactification, by extended weighted tropical curves.”
The tropicalization of the moduli space of curves (2012, unknown): “Similar objects were named stacky fans in [BMV11, CMV12], but that term is also standard for combinatorial data associated to toric stacks [BCS05].”
Tropicalizing the space of admissible covers (2014, background): “The moduli space of genus g tropical curves, roughly speaking, parametrizes vertex weighted metric graphs of a given genus [8, 10, 13].”
A MODULI STACK OF TROPICAL CURVES (2017, unknown): “ry(see, e.g., [Cap13], [CMV13], [Cap16], and [Cha16] for survey papers). In particular, we refer the reader to [Mik06], [Mik07], [GKM09], and [GM08] for the case of genus g = 0, as well as to [CV10], [BMV11], [Cha12], and [Viv13] for the higher genus case and the connections to the tropical Torelli ”
Degeneration of Linear Series from the Tropical Point of View and Applications (2015, background): “dimension g, taking an abelian variety to its skeleton in the sense of Berkovich. Finally, there are Torelli maps M g !A g (resp. Mtrop !Atrop) which take a curve (resp. metric graph) to its Jacobian [BMV11]. Theorem 6.5 implies that the following square commutes: M g trop / Mtrop A g trop /Atrop ”
Tropical hyperelliptic curves (2011, methodology): “s together two strands in the recent tropical geometry literature: tropicalBrill-Noether theory on the one hand [Bak08], [Cap11b], [CDPR10], [LPP11]; and tropical moduli spaces of curves on the other [BMV11], [Cap10], [Cap11a], [Cha11], [MZ07]. The work of Baker and Norine in [BN07] and Baker in [Ba”
Riemann–Roch theory for weighted graphs and tropical curves (2011, methodology): “r the specialization. Such metric weighted graphs are called weighted tropical curves; they admit a moduli space, Mtrop g , whose topological properties have strong similarities with those of Mg; see [BMV] and [C2]. The connections between the algebraic and the tropical theory of curves have been th”
Combinatorics of the tropical Torelli map (2010, methodology): “Our construction is different from the one in [BMV 2011], though it is still very much inspired by the ideas in that paper.”
Gonality of algebraic curves and graphs (2012, result): “some closely related issues are currently being investigated, under a completely diﬀerent perspective, in a joint project of O. Amini, M. Baker, E. Brugall´e and J. Rabinoﬀ. We refer also to [BPR11], [BMV11], [C11], [Ch11] and [LPP] for some recent work on the relation between algebraic and tropical”
Moduli of tropical plane curves (2014, methodology): “We refer to [7,11] for the definition ofMg , combinatorial details, and applications in algebraic geometry.”