Tautological classes on the moduli spaces of stable maps to Pr via torus actions

We present a localization proof of the fact that the cohomology of the moduli spaces of genus zero stable maps to projective spaces is entirely tautological. In addition, we obtain a description of a Bialynicki-Birula decomposition of the stack of stable maps in the context of Gathmann's moduli spaces of maps with prescribed contact order to a fixed hyperplane. We show that the decomposition is filterable.