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The Moduli Space of Curves, Double Hurwitz Numbers, and Faber’s Intersection Number Conjecture
I. P. Goulden1, D. M. Jackson1 and R. Vakil2
1Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
{ipgoulden, dmjackson}@math.uwaterloo.ca
2Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, CA 94305, USA
vakil@math.stanford.edu
Annals of Combinatorics 15 (3) pp.381-436 July, 2011
AMS Subject Classification: 14H10, 05E99, 14K30
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 P1 with given ramification over ∞ 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 ψ-classes. As genus 0 double Hurwitz numbers with at most 3 parts over ∞ 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 κg−2.
Keywords: moduli space of curves, branched covers, Chow rings, enumerative geometry

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