Chow Rings of Mp₀,₂(N,d) and Mbar₀,₂(Pⁿ⁻¹,d) and Gromov-Witten Invariants of Projective Hypersurfaces of Degree 1 and 2

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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 Mbar₀,₂(Pⁿ⁻¹,d), the moduli spaces of stable maps from genus 0 stable curves to projective space Pⁿ⁻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 Mp₀,₂(N,d), the moduli space of quasi maps from P¹ to Pⁿ⁻¹ with two marked points. In order to prove our formulas, we use the results on Chow ring of Mbar₀,₂(Pⁿ⁻¹,d), that were derived by A. Mustata and M. Mustata. We also present explicit toric data of Mp₀,₂(N,d) and prove relations of Chow ring of Mp₀,₂(N,d).