A MODULI STACK OF TROPICAL CURVES
By Renzo Cavalieri, Melody Chan, Martin Ulirsch, Jonathan Wise. Forum of Mathematics, Sigma, 2017
94 citations recorded by the source corpus.
We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves. Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/91bb4bcda91267654f01ff0586b811194bad72a5
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In Algebra and Number Theory
- Dhruv Ranganathan
- D. Abramovich
- Qile Chen
- Mark Gross
- Bernd S Siebert
- Jonathan Wise
- Martin Ulirsch
- Helge Ruddat
In Computational Theory and Mathematics
- Dhruv Ranganathan
- Mark Gross
- Bernd S Siebert
- D. Abramovich
- Qile Chen
- P. Hacking
- Markus Reineke
- Martin Ulirsch
In Geometry and Topology
- Dhruv Ranganathan
- D. Abramovich
- Mark Gross
- Bernd S Siebert
- Qile Chen
- Jonathan Wise
- Helge Ruddat
- Martin Ulirsch
Citations in the corpus
- Logarithmic Gromov–Witten theory with expansions (2019, methodology): “We will not need to make the rational equivalence here precise, but if we use the categorical equivalence of cone complexes with Artin fans, a precise translation can be made in the Chow group of an appropriate family of Artin stacks [16, 74], and this implies the statement in [4, Section 4.”
- Decomposition of degenerate Gromov–Witten invariants (2017, background): “iven by multiplication by a non-negative integer m ˝. (2)The multiplicity of the divisor ˇ(f0g) along D ˝ is m ˝: In other words, we have an equality of Weil divisors (3.1.1) ˇ(f0g) = X ˝ m ˝D ˝: 6In [CCUW17], Chan, Cavalieri, Ulirsch and Wise redene moduli of tropical curves as stacks, so this issu”