Tropicalizing the space of admissible covers
By Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan. 2014
98 citations recorded by the source corpus.
We study the relationship between tropical and classical Hurwitz moduli spaces. Following recent work of Abramovich, Caporaso and Payne, we outline a tropicalization for the moduli space of generalized Hurwitz covers of an arbitrary genus curve. Our approach is to appeal to the geometry of admissible covers, which compactify the Hurwitz scheme. We study the relationship between a combinatorial moduli space of tropical admissible covers and the skeleton of the Berkovich analytification of the classical space of admissible covers. We use techniques from non-archimedean geometry to show that the tropical and classical tautological maps are compatible via tropicalization, and that the degree of the classical branch map can be recovered from the tropical side. As a consequence, we obtain a proof, at the level of moduli spaces, of the equality of classical and tropical Hurwitz numbers.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/5c9042caa484ffaa61e48cb8d7c0e02454346f91
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Algebra and Number Theory
In Computational Theory and Mathematics
In Geometry and Topology
Citations in the corpus
- Brill-Noether theory for curves of a fixed gonality (2017, background): “From the results of [16], the tropicalization of Mg coincides with the image of the tropical Hurwitz space of simply ramified degree k covers of P1 in M trop g .”
- Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, result): “Similar results have appeared for other moduli spaces, such as the moduli space of admissible covers (see [11]), the moduli space of weighted stable curves (see [10] for the case of genus g = 0 and [50] for g 0), and the moduli space of rational logarithmic stable maps into a toric variety (see [45]”
- Skeletons and Fans of Logarithmic Structures (2015, unknown): “e of the primary applications of tropical geometry, and therefore of fans, is through moduli spaces. Mikhalkin’s correspondence theorem [Mik05], Nishinou and Siebert’s vast extension [NS06], the work [CMR] of Cavalieri, Markwig and Ranganathan, and the manuscript [ACGS13], all show that tropical mod”