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

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 the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i - g + 1 genus 0 components. As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Diaz, Faber, Getzler, Ionel, Looijenga, Pandharipande, and others). This theorem appears to be the geometric content behind these results; the rest is straightforward graph combinatorics. The theorem also suggests the importance of the stratification of the moduli space by the number of rational components.