2 Fe b 19 98 INTERSECTION NUMBERS ON THE MODULI SPACES OF STABLE MAPS IN GENUS 0

Let V be a smooth, projective, convex variety. We define tautological ψ and κ classes on the moduli space of stable maps M0,n(V ), give a (graphical) presentation for these classes in terms of boundary strata, derive differential equations for the generating functions of the Gromov-Witten invariants of V twisted by these tautological classes, and prove that these intersection numbers are completely determined by the Gromov-Witten invariants of V . This results in families of Frobenius manifold structures on the cohomology ring of V which includes the quantum cohomology as a special case.