  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}.
