Intersections of Q-Divisors on Kontsevich's Moduli Space $\bar{M}_{0,n}(P^r,d)$ and Enumerative Geometry

We study the geometry of varieties parametrizing degree d rational and elliptic curves in P^n intersecting fixed general linear spaces and tangent to a fixed hyperplane H with fixed multiplicities along fixed general linear subspaces of H. As an application, we derive recursive formulas for the number of such curves when the number is finite. These recursive formulas require as ``seed data'' only one input: there is one line in P^1 through two points. These numbers can be see...

This paper is an elementary introduction to the theory of moduli spaces of curves and maps. As an application to enumerative geometry, we show how to count the number of bitangent lines to a projective plane curve of degree $d$ by doing intersection theory on moduli spaces.

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 ar...

We use twisted stable maps to answer the following question. Let E\subset P^2 be a smooth cubic. How many rational degree d curves pass through a general points of E, have b specified tangencies with E and c unspecified tangencies, and pass through 3d-1-a-2b-c general points of P^2? The answer is given as a generalization of Kontsevich's recursion. We also investigate more general enumerative problems of this sort, and prove an analogue of a formula of Caporaso and Harris.

We begin a study of the intersection theory of the moduli spaces of degree two stable maps from two-pointed rational curves to arbitrary-dimensional projective space. First we compute the Betti numbers of these spaces using Serre polynomial and equivariant Serre polynomial methods developed by E. Getzler and R. Pandharipande. Then, via the excision sequence, we compute an additive basis for their Chow rings in terms of Chow rings of nonlinear Grassmannians, which have been de...

On a stack of stable maps, the psi classes are modified by subtracting certain boundary divisors. These modified psi classes are compatible with forgetful morphisms, and are well-suited to enumerative geometry: tangency conditions allow simple expressions in terms of modified psi classes. Topological recursion relations are established among their top products in genus zero, yielding effective recursions for characteristic numbers of rational curves in any projective homogene...

We describe algorithms for computing the intersection numbers of divisors and of Chern classes of the Hodge bundle on the moduli spaces of stable pointed curves. We also discuss the implementations and the results obtained. There are several applications. We discuss one in particular: the calculation of the projection in the tautological ring of the moduli space of abelian varieties of the class of the locus of Jacobians.

We consider the cones of curves and divisors on the moduli space of stable pointed rational curves,M_n, and on the quotient by the symmetric group, Q_n, which is a moduli space of pairs. We find generators for contractible extremal rays of the cone of curves NE_1(M_n), and for the cone of divisors NE^1(Q_n). This second cone turns out to be simplicial. We give complete descriptions of NE_1(M_n) and NE_1(Q_n) for small n (< 8 in the first case, < 11 in the second). We also hav...

We investigate the spaces of rational curves on a general hypersurface. In particular, we show that for a general degree $d$ hypersurface in $\mathbb{P}^n$ with $n \geq d+2$, the space $\overline{\mathcal{M}_{0,0}}(X,e)$ of degree $e$ Kontsevich stable maps from a rational curve to $X$ is an irreducible local complete intersection stack of dimension $e(n-d+1)+n-4$.

In this thesis I give a new description for the moduli space of stable n pointed curves of genus zero and explicitly specify a natural isomorphism and inverse between them that preserves many important properties. I also give a natural description for the universal curve of this space. These descriptions are explicit and defined in a straight forward way. I also compute the tangent bundle of this space. In the second part of the thesis I compute the ordinary integral cohomolo...