2006/11/21 by I. P. Goulden, Ian P. Goulden, Goulden, Ian P. +5 · 2 citations
Computer Science · Mathematics · #14K30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Primary 14H10 #Secondary 05E99 #math.AG #math.CO #msc:05E99 #msc:14H10 #msc:14K30
paper · pdf · doi:10.48550/arxiv.math/0611659
45 pages, 6 figures
arxiv created 2006/11/21 · openalex publication_date 2006/11/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 kappag-2.