[Paper Review] Tropicalizing the space of admissible covers

Renzo Cavalieri, Hannah Markwig+1 more|arXiv (Cornell University)|Jan 19, 2014

Polynomial and algebraic computation22 references4 citations

TL;DR

This paper establishes a geometric and functorial correspondence between classical and tropical Hurwitz moduli spaces via tropicalization, showing that the classical branch map degree—encoding Hurwitz numbers—can be recovered from tropical data. By analyzing the skeleton of the Berkovich analytification of admissible covers and proving that tropicalization commutes with tautological maps, the authors reprove the equality of classical and tropical Hurwitz numbers at the moduli space level using non-archimedean geometry and deformation theory.

ABSTRACT

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 define and construct a moduli space of tropical admissible covers, and study its relationship with 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.

Motivation & Objective

• To establish a geometric and functorial relationship between classical and tropical Hurwitz moduli spaces.
• To show that tropicalization commutes with the source and branch maps on these moduli spaces.
• To recover the degree of the classical branch map from the tropical side, thereby proving the equality of classical and tropical Hurwitz numbers.

Proposed method

• The authors use the Berkovich analytification of the moduli space of admissible covers and study its skeleton as a cone complex.
• They define a tropicalization map from the analytification to the moduli space of tropical admissible covers.
• They prove that this map factors through the skeleton and induces a surjective face morphism of cone complexes.
• They compute the dilation factor of the tropical branch map using deformation theory and valuation theory of smoothing parameters.
• They define weights on combinatorial types of tropical covers as products of local Hurwitz numbers, expansion factors, and automorphism factors.
• They show that the classical Hurwitz number equals the weighted sum of these tropical contributions over all combinatorial types lying over a fixed target tropical curve.

Experimental results

Research questions

• RQ1 How does the tropicalization of the moduli space of admissible covers relate to the skeleton of its Berkovich analytification?
• RQ2 Does tropicalization commute with the tautological source and branch maps on the moduli spaces of admissible covers?
• RQ3 Can the degree of the classical branch map, which computes Hurwitz numbers, be recovered from tropical data?
• RQ4 What is the precise relationship between the integral structures of the cone complexes on the classical and tropical sides?
• RQ5 How do deformation parameters and valuations of smoothing parameters determine the tropicalization map?

Key findings

• The tropicalization map from the analytification of the moduli space of admissible covers factors through its skeleton and induces a surjective face morphism of cone complexes.
• The tropical branch map is a morphism of cone complexes with integral structure, and its dilation factor matches the local Hurwitz number contribution for each combinatorial type.
• The classical Hurwitz number equals the sum over all tropical combinatorial types of the product of the weight ω(Θ) and the dilation factor dΘ(brtrop).
• For double Hurwitz numbers (h=0 with two non-simple ramification types), the formula reduces to a sum over trivalent target curves weighted by 1/|Aut(Γ)| and products of edge weights.
• The total degree of the branch map is computed as the sum of contributions from all top-dimensional cells in the tropical moduli space, with each contribution determined by the LCM of ramification indices and automorphism factors.
• The results confirm the classical correspondence theorem for Hurwitz numbers at the level of moduli spaces, using tropical geometry and non-archimedean techniques.