O ct 2 00 4 INTERMEDIATE MODULI SPACES OF STABLE MAPS

We describe the Chow ring with rational coefficients of M0,1(P , d) as the subring of invariants of a ring B(M0,1(P , d);Q), relative to the action of the group of symmetries Sd. We compute B(M0,1(P , d);Q) by following a sequence of intermediate spaces for M0,1(P , d). Introduction The moduli spaces of stable maps from curves to smooth projective varieties were introduced by M.Kontsevich and Y.Manin in [KM]. They provided the set-up for an axiomatic algebro-geometric approach to GromovWitten theory, generating beautiful results in enumerative geometry and mirror symmetry. Gromov-Witten invariants, defined as intersection numbers on the moduli spaces of stable maps, were computed by recurrence methods. An important role in these methods was played by the “boundary divisors” of the moduli space, parametrizing maps with reducible domains. In the case when the domain curve is rational and the target is Pn, the functor M0,m(P n, d) is represented by a smooth Deligne-Mumford stack, M0,m(P n, d). Here the generic member is a smooth, degree d, rational curve in Pn with m distinct marked points. The boundary is made of degree dmorphisms μ : C → Pn from nodal m-pointed curves C of arithmetic genus 0, such that every contracted component of C has at least 3 special points: some of the m marked points or nodes. The cohomology ring of M0,m(P n, d) is not known in general. K.Behrend and A.O’Halloran in [BO] have outlined an approach for computing the cohomology ring for m = 0. They rely on a method of Akildiz and Carell, applied to a C∗-equivariant vector field on M0,0(P n, d). They give a complete set of generators and relations for the case d = 2 and for the ring of M0,0(P ∞, 3). The main result of this paper is a description of the Chow ring with rational coefficients of M0,1(P n, d). Our method is different from the one employed by K.Behrend and A.O’Halloran, relying on a sequence of intermediate moduli spaces. In the main theorem (Theorem 4.25), we express A∗(M 0,1(P n, d);Q) as the subring of invariants of a ring B(M0,1(P n, d);Q), relative to the action of the group of symmetries Sd. We give a complete set Date: February 26, 2008. 1 2 ANDREI MUSTAŢǍ AND MAGDALENA ANCA MUSTAŢǍ of generators and relations for B∗(M 0,1(P n, d);Q), the geometric significance of which will be explained here in more detail. Motivated by results in mirror symmetry, Givental in [G], and Lian, Liu and Yau in [LLY] have computed Gromow-Witten invariants for hypersurfaces in Pn using the Bott residue formula and the existence of a birational morphism φ : M0,0((P n × P), (d, 1)) → Pd . Here Pd := P (n+1)(d+1)−1 parametrizes (n+ 1) degree dpolynomials in one variable, modulo multiplication by constants. Following Givental, we will call the domain of φ the graph space. We will use the short notation G(Pn, d) for it. Of the various boundary divisors of G(Pn, d) and their images in Pd , the most notable for us is M0,1(P n, d) × P1, mapped by φ into Pn × P1. The productM0,1(P n, d)×P1 is embedded in G(Pn, d) as the space parametrizing splitted curves C1 ∪ C2, were C1 comes with a degree (d, 0) morphism to Pn×P1 and C2 comes with a degree (0, 1) morphism. Our study starts from the diagram M 0,1(P n, d)× P1 // G(Pn, d)