Chow rings of <(Mp)over tilde>(0,2) (N, d) and <(M)over bar>(0,2)(P-N (-1), d) and Gromov-Witten invariants of projective hypersurfaces of degree 1 and 2

摘要

In this paper, we prove formulas that represent two-pointed Gromov-Witten invariant < OhaOhb >(0,d) of projective hypersurfaces with d = 1, 2 in terms of Chow ring of (M) over bar (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(OhaOhb)(0,d), which was introduced by Jinzenji, in terms of Chow ring of (Mp) over tilde (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 (M) over bar (0,2)(P-N (-) (1), d), that were derived by Mustata and Mustata. We also present explicit toric data of (Mp) over tilde (0,2)(N, d) and prove relations of Chow ring of (Mp) over tilde (0,2)(N, d).