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.
ABSTRACT. We study the Berkovich analytification of the space of genus 0 logarithmic stable maps to a toric variety X and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two new geometric descriptions of this space of maps - (1) as an explicit toroidal modification of M 0,n ×X and (2) as a tropical compactification in a toric variety. On the combinatorial side, we prove that the tropicalization of the space of genus 0 logarithmic stable maps coincides with the space of tropical stable maps, giving a large new collection of examples of faithful tropicalizations for moduli. Moreover, we identify the optimal settings in which the tropicalization of the moduli space of maps is faithful. The Nishinou-Siebert correspondence theorem is shown to be a consequence of this geometric connection between the algebraic and tropical moduli.
We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector w of weights, the moduli space of tropical w-stable curves can be given the structure of a balanced fan if and only if w has only heavy and light entries. In this case, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.
We construct a version of relative Gromov-Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair (X, D), we construct virtually smooth and proper moduli spaces of curves in X with prescribed boundary conditions along D. Each point in such a moduli space parameterizes maps from nodal curves to expanded degenerations of X that are dimensionally transverse to the strata. We use the expanded formalism to reconstruct the virtual class attached to a tropical map in terms of spaces of maps to expansions attached to the vertices.
This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus 1. We construct a smooth and proper moduli space dominating the main component of Kontsevich's space of stable genus 1 maps to projective space. A variation on this theme furnishes a modular interpretation for Vakil and Zinger's famous desingularization of the Kontsevich space of maps in genus 1. Our methods also lead to smooth and proper moduli spaces of pointed genus 1 quasimaps to projective space. Finally, we present an application to the log minimal model program for M1,n. We construct explicit factorizations of the rational maps among Smyth's modular compactifications of pointed elliptic curves.
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