A presentation for the Chow ring A^*(\bar{M}_{0,2}(P^1,2))

Abstract

The purpose of this dissertation is to study the intersection theory of the moduli spaces of stable maps of degree two from two-pointed, genus zero nodal curves to arbitrary-dimensional projective space. Toward this end, first the Betti numbers of \bar{M}_{0,2}(P^r,2) are computed using Serre polynomials and equivariant Serre polynomials. Then, specializing to the space \bar{M}_{0,2}(P^1,2), generators and relations for the Chow ring are given. Chow rings of simpler spaces are also described, and the method of localization and linear algebra is developed. Both tools are used in finding the relations. It is further demonstrated that no additional relations exist among the generators, so that a presentation for the Chow ring A^*(\bar{M}_{0,2}(P^1,2)) is obtained. As a further check of the presentation, it is applied to give a new computation of the previously known genus zero, degree two, two-pointed gravitational correlators of P^1. Portions of this work also appear in math.AG/0501322 and math.AG/0504575, but the dissertation contains significantly more background and detail for those who may be interested in these. The dissertation is preserved in original form except for spacing changes, elimination of some front and end materials, additions to some references, and correction of typos in Proposition 11.