  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. (The proof of this result is
quite short and naive.) As consequences, we prove and generalize various
conjectures and theorems about various moduli spaces of curves (due to Getzler,
Ionel, Faber, Looijenga, Pandharipande, Diaz, 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 number of rational components.
