The tautological classes of the moduli spaces of stable maps to flag varieties

Abstract

We study the tautological classes of the Kontsevich-Manin moduli spaces of genus 0 stable maps to SL flag varieties. We prove that the rational cohomology and rational Chow rings of these spaces are isomorphic and that they are generated by tautological classes. In the case when the target is a projective space, we present a second proof of this result in the spirit of Gromov-Witten theory by making use of a suitable torus action. In addition, we explicitly describe a Bialynicki-Birula stratification of the Kontsevich-Manin spaces in terms of the Gathmann-Li spaces of relative stable morphisms. Finally, we analyze the small codimension classes on the space of maps to arbitrary flag varieties. We obtain an explicit description of the Picard groups. We formulate a conjecture about relations between the tautological generators, which we check in low codimension.