Introduction

Abstract

The aim of enumerative geometry is to count how many geometric figures satisfy given conditions. The most basic example is the question, How many lines are there through two distinct points? A natural extension of this question is the problem of determining the number N _d of rational curves of degree d passing through 3d - 1 points in general position in the complex projective plane. The number 3J - 1 is not arbitrary: it matches the dimension of the family of curves under consideration, so it is precisely the right number of conditions to impose in order to get a finite number of solutions.