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We prove a localization formula for the moduli space of stable relative maps.As an application, we prove that all codimension i tautological classes on themoduli space of stable pointed curves vanish away from strata corresponding tocurves with at least i-g+1 genus 0 components. (The proof of this result isquite short and naive.) As consequences, we prove and generalize variousconjectures and theorems about various moduli spaces of curves (due to Getzler,Ionel, Faber, Looijenga, Pandharipande, Diaz, and others). This theorem appearsto be the geometric content behind these results; the rest is straightforwardgraph combinatorics. The theorem also suggests the importance of thestratification of the moduli space by number of rational components.

Relative virtual localization and vanishing of tautological classes on moduli spaces of curves