The enumerative geometry of rational and elliptic curves in projective space

TL;DR

This paper develops recursive formulas to count rational and elliptic curves in projective space intersecting fixed linear spaces, providing insights into the topology of related moduli spaces and offering an algorithm for such enumerations.

Contribution

It introduces recursive formulas for enumerating rational and elliptic curves in projective space, linking intersection theory with the Chow ring of moduli spaces.

Findings

Recursive formulas for counting curves with fixed intersection conditions.

Connection of enumerative counts to the Chow ring and topology of moduli spaces.

Algorithm for counting rational and elliptic curves in projective space.

Abstract

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 seen as top intersection products of various cycles on the Hilbert scheme of degree d rational or elliptic curves in P^n, or on certain components of $\mbar_0(P^n,d)$ or $\mbar_1(P^n,d)$, and as such give information about the Chow ring (and hence the topology) of these objects. The formula can also be interpreted as an equality in the Chow ring (not necessarily at the top level) of the appropriate Hilbert scheme or space of stable maps. In particular, this gives an algorithm for counting rational and elliptic curves in P^n intersecting various fixed general linear spaces. (The genus 0 numbers were found earlier by Kontsevich-Manin, and the genus 1 numbers were found for n=2 by Ran and Caporaso-Harris, and independently by Getzler for n=3.)