Enumerative Geometry of Hyperelliptic Plane Curves

In recent years there has been a tremendous amount of progress on classical problems in enumerative geometry. This has largely been a result of new ideas and motivation for these problems coming from theoretical physics. In particular, the theory of Gromov-Witten invariants has provided powerful tools for counting curves satisfying incidence conditions. This theory has been most successful in dealing with questions about rational curves. This is partly because it is much more common for the genus 0 invariants to correspond to enumerative problems. In addition, it is much easier to compute these invariants. This is due mainly to the existence of the WDVV equations. There has been success in extending the techniques used to derive these equations to find recursions satisfied by the invariants in genus 1 and 2. ([G],[BP]). In some situations these higher genus invariants also correspond to classical enumerative problems. Thus, the theory gives new methods to solve these problems. We will explore a different approach to using Gromov-Witten theory to solve enumerative problems involving higher genus curves. Rather than generalizing the methods that succeed in genus 0, we try to reduce questions in higher genus to questions about rational curves. We utilize the well-developed theory of genus 0 Gromov-Witten invariants to solve enumerative problems involving hyperelliptic curves in P. Our main enumerative result is the construction of a recursive algorithm which counts the number of hyperelliptic plane curves of degree d and genus g passing through 3d+ 1 general points. The basic facts about stable maps and Gromov-Witten theory are reviewed in Section 1. The main results we need are the WDVV equations which allow us to compute, and an explicit representation of the virtual fundamental class as a Chern class. In Section 2 we introduce the main idea of the paper. Thinking of a map from a hyperelliptic curve as a family of maps from pairs of points parametrized by P gives us a natural correspondence between