TAUTOLOGICAL CLASSES ON THE MODULI SPACES OF STABLE MAPS TO Pr VIA TORUS ACTIONS

In our previous paper [18], we introduced the tautological rings of the genus zero moduli spaces of stable maps to homogeneous spaces X. We showed that in the case of SL flags, all rational cohomology classes on the stable map spaces are tautological using methods from Hodge theory. The purpose of this note is to indicate a localization proof, in the spirit of Gromov-Witten theory, when X is a projective space. To set the stage, we recall the definition of the tautological rings. The moduli stacks M0,S(P r, d) parametrize S-pointed, genus 0, degree d stable maps to Pr. We use the notation M0,n(P r, d) when the labeling set is S = {1, 2, . . . , n}. These moduli spaces are connected by a complicated system of natural morphisms, which we enumerate below: • forgetful morphisms: π : M0,S(P r, d) → M0,T (P r, d) defined for T ⊂ S. • gluing morphisms which produce maps with nodal domains,