Recently, there has been great interest in the application of composition
laws to problems in enumerative geometry. Using the moduli space of stable
maps, we compute the number of irreducible, reduced, nodal, degree-$d$
genus-$2$ plane curves whose normalization has a fixed complex structure and
which pass through $3d - 2$ general points in $\Bbb P^2$.Comment: 13 pages, AMS-Te

Katz, Sheldon

Qin, Zhenbo

Ruan, Yongbin

English

arXiv

this paper is to compute N 2;d for d  4 (see (1.1)). The lower degree cases for N g;d were known classically for many years. Not much progress has been made until the introduction of quantum cohomology theory. Based on ideas from physics, a recursion formula for N 0;d was proved by [R-T, K-M]. The influence of physical ideas opens up entirely new directions in enumerative geometry. Roughly speaking, N 0;d can be interpreted as a correlation function in a certain topological quantum field theory (topological sigma model [Wit]). All topological quantum field theories have a composition law, which in this instance gives the beautiful recursion formula for N 0;d . Furthermore, the composition law naively suggests a recursion formula for N g;d in terms of N 0;d . Unfortunately, a simple calculation of lower degree elliptic curves showed that the formula from physics always gives a wrong answer for higher genus case g ? 0. It was showed in [R-T] that the correlation function \Psi g;d (Gromov-Witten invariants) of the topological sigma model counts the number of perturbed pseudo-holomorphic maps. Moreover, it satisfies the composition law physicists predicted and hence can be computed b

Composition law and Nodal genus-2 curves in P^2

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