Chow Rings of Mp_{0,2}(N,d) and Mbar_{0,2}(P^{N-1},d) and Gromov-Witten Invariants of Projective Hypersurfaces of Degree 1 and 2

TL;DR

This paper derives formulas for Gromov-Witten invariants of degree 1 and 2 projective hypersurfaces using Chow rings of moduli spaces of stable maps and quasi-maps, connecting intersection theory with enumerative geometry.

Contribution

It introduces explicit formulas relating Gromov-Witten invariants to Chow rings of moduli spaces, utilizing and extending previous Chow ring results for these spaces.

Findings

Formulas for Gromov-Witten invariants in terms of Chow rings.

Explicit toric data for moduli space Mp_{0,2}(N,d).

Relations of Chow ring of Mp_{0,2}(N,d) established.

Abstract

In this paper, we prove formulas that represent two-pointed Gromov-Witten invariant <O_{h^a}O_{h^b}>_{0,d} of projective hypersurfaces with d=1,2 in terms of Chow ring of Mbar_{0,2}(P^{N-1},d), the moduli spaces of stable maps from genus 0 stable curves to projective space P^{N-1}. Our formulas are based on representation of the intersection number w(O_{h^a}O_{h^b})_{0,d}, which was introduced by Jinzenji, in terms of Chow ring of Mp_{0,2}(N,d), the moduli space of quasi maps from P^1 to P^{N-1} with two marked points. In order to prove our formulas, we use the results on Chow ring of Mbar_{0,2}(P^{N-1},d), that were derived by A. Mustata and M. Mustata. We also present explicit toric data of Mp_{0,2}(N,d) and prove relations of Chow ring of Mp_{0,2}(N,d).