Notes on stable maps and quantum cohomology

Abstract

Contents 0. Introduction 1 1. Stable maps and their moduli spaces 10 2. Boundedness and a quotient approach 12 5. The construction of M g;n (X; fi) 25 6. The boundary of M 0;n (X; fi) 29 7. Gromov-Witten invariants 31 8. Quantum cohomology 34 9. Applications to enumerative geometry 38 10. Variations 43 References 46 0. Introduction 0.1. Overview. The aim of these notes is to describe an exciting chapter in the recent development of quantum cohomology. Guided by ideas from physics (see [W]), a remarkable structure on the solutions of certain rational enumerative geometry problems has been found: the solutions are coefficients in the multiplication table of a quantum cohomology ring. Associativity of the ring yields non-trivial relations among the enumerative solutions. In many cases, these relations suffice to solve the enumerative problem. For example, let N d be the number of degree d, rational plane curves passing through 3d \\Gamma 1 general points in P . Since there is a un