Tautological classes on the moduli spaces of stable maps to projective   spaces
  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 stratification in the context of Gathmann's moduli spaces of maps with prescribed contact order to a fixed hyperplane. 
Dragos Oprea