The enumerative geometry of rational and elliptic curves in projective space

摘要

We study the geometry of moduli spaces of genus 0 and 1 curves in P-n with specified contact with a hyperplane H. We compute intersection numbers on these spaces that correspond to the number of degree d curves incident to various general linear spaces, and tangent to H with various multiplicities along various general linear subspaces of H. (The numbers of classical interest, the numbers of curves incident to various general linear spaces and no specified contact with H, are a special case.) In the genus 0 case, these numbers are candidates for relative Gromov-Witten invariants of the pair (P-n, H), and in the genus I case they generalize the enumerative consequences of Kontsevich's reconstruction theorem for P-n. The intersection numbers are recursively computed by degenerating conditions. As an example, the enumerative geometry of quartic elliptic space curves is worked out in detail. [References: 32]