| The purpose of this dissertation is to study the intersection theory of the moduli spaces M0,2 ( Pr , 2) of stable maps of degree two from two-pointed, genus zero nodal curves to arbitrary-dimensional projective space. Moduli spaces of stable maps were introduced by Kontsevich in 1994 and have proven useful in studying both superstring theory and enumerative geometry. The intersection theory of M0,2 ( Pr , 2) can be completely described by giving a presentation for its Chow ring. Toward this end, first the Betti numbers of M0,2 ( Pr , 2) are computed using Serre polynomials and equivariant Serre polynomials. Then, specializing to the space M0,2 ( P1 , 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*( M0,2 ( P1 , 2)) is obtained. This presentation is the main result of the dissertation, and is given by A* &parl0;M0,2 &parl0;P1,2&parr0;&parr0; ≃ Q D0,D1 ,D2,H1 ,H2,y 1,y2 H21 ,H22,D0y1, D0y2,D2-y 1-y2,y1-1 4D1-14D 2-D0+H1,y2 -14D1-1 4D2-D0+H2, D1+D23, D1y1y2 . This is the first known presentation for a Chow ring of a moduli space of stable maps of degree greater than one with more than one marked point. 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 P1 . Graduate level knowledge of algebraic geometry and intersection theory are prerequisites for understanding this dissertation. Basic knowledge of stacks is also assumed, but most of the dissertation can be understood without it. |
