The moduli space of curves, double Hurwitz numbers, and Faber's intersection number conjecture

Abstract

We define the dimension 2g-1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P^1 with given ramification over infinity and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localization expresses such classes in terms of localization trees weighted by ``top intersections'' of tautological classes and genus 0 double Hurwitz numbers. This identity of generating series can be inverted, yielding a ``combinatorialization'' of top intersections of psi-classes. As genus 0 double Hurwitz numbers with at most 3 parts over infinity are well understood, we obtain Faber's Intersection Number Conjecture for up to 3 parts, and an approach to the Conjecture in general (bypassing the Virasoro Conjecture). We also recover other geometric results in a unified manner, including Looijenga's theorem, the socle theorem for curves with rational tails, and the hyperelliptic locus in terms of kappa_{g-2}.

45 pages, 6 figures

References in corpus (2)

• Three questions in Gromov-Witten theory
• Generating Functions for Hurwitz-Hodge Integrals

Cited by in corpus (5)

• Gerby Localization, Z_3-Hodge Integrals and the GW Theory of C^3/Z_3
• A short proof of the lambda_g-conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves
• Polynomial families of tautological classes on ${\mathcal{M}}_{g,n}^{rt}$
• On independence of generators of the tautological rings
• Tropical Hurwitz Spaces