The tropicalization of the moduli space of curves
By D. Abramovich, Lucia Caporaso, Sam Payne. 2012
231 citations recorded by the source corpus.
We show that the skeleton of the Deligne-Mumford-Knudsen moduli stack of stable curves is naturally identified with the moduli space of extended tropical curves, and that this is compatible with the "naive" set-theoretic tropicalization map. The proof passes through general structure results on the skeleton of a toroidal Deligne-Mumford stack. Furthermore, we construct tautological forgetful, clutching, and gluing maps between moduli spaces of extended tropical curves and show that they are compatible with the analogous tautological maps in the algebraic setting.
Subfields: Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/35f2dd4e1ec8270e907f7e06bff92f8c7d4fb470
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In Computational Theory and Mathematics
- Dhruv Ranganathan
- Martin Ulirsch
- Michael Joswig
- David Jensen
- Renzo Cavalieri
- Hannah Markwig
- Sam Payne
- Simon Hampe
In Geometry and Topology
- Dhruv Ranganathan
- Martin Ulirsch
- David Jensen
- Michael Joswig
- Sam Payne
- Renzo Cavalieri
- Hannah Markwig
- Bernd Sturmfels
Citations in the corpus
- MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY (2014, methodology): “4, we explore the connection with Berkovich skeletons, in the spirit of the results obtained in [1, 26].”
- Logarithmic Gromov–Witten theory with expansions (2019, background): “After a polyhedral subdivision of ΣY [1], this rigid tropical map can be made transverse – i.”
- Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, background): “In [1], as an archetypical result, the authors show that the moduli space of tropical curves is isomorphic to the skeleton of the moduli space of algebraic curves with the respect to the toroidal structure coming from the Deligne–Knudsen–Mumford compactification (see [14] and [34]).”
- Moduli of tropical plane curves (2014, background): “The map (Equation 10) is referred to as ‘naive set-theoretic tropicalization’ by Abramovich, Caporaso, and Payne [1].”
- Brill-Noether theory for curves of a fixed gonality (2017, background): “This is precisely the situation for the moduli space of curves [1].”
- Tropical independence I: Shapes of divisors and a proof of the Gieseker–Petri theorem (2014, background): “This follows from the fact that the moduli space of tropical curves is the skeleton of the Deligne–Mumford compactification of the moduli space of curves [Abramovich et al. 2012], and can also be proved by deformation theory, as in [Baker 2008, Appendix B].”