  We prove the genus zero part of the generalized Witten conjecture relating
moduli spaces of spin curves to Gelfand-Dickey hierarchies. That is, we show
that intersection numbers on the moduli space of stable r-spin curves assemble
into a generating function which yields a solution of the semiclassical limit
of the KdV_r equations. We formulate axioms for a cohomology class on this
moduli space which allow one to construct a cohomological field theory of rank
$r-1$ in all genera.
  In genus zero it produces a Frobenius manifold which is isomorphic to the
Frobenius manifold structure on the base of the versal deformation of the
A_{r-1} singularity. We prove analogs of the puncture, dilaton, and topological
recursion relations by drawing an analogy with the construction of
Gromov-Witten invariants and quantum cohomology.

Algebraic Geometry;Differential Geometry;Quantum Algebra