Moduli Space of Quasi-Maps from P^{1} with Two Marked Points to P(1,1,1,3) and j-invariant

TL;DR

This paper constructs the toric structure of the moduli space of degree d quasi-maps from P^1 with two marked points to P(1,1,1,3), proving its compactness, describing its Chow ring, and confirming a conjecture relating intersection numbers to the inverse j-invariant series.

Contribution

It provides the explicit toric data for the moduli space, proves its compactness as an orbifold, and verifies Jinzenji's conjecture connecting intersection theory with the j-invariant.

Findings

The moduli space is a compact toric orbifold.

Chow ring of the moduli space is explicitly determined.

Intersection numbers match inverse j-invariant series coefficients.

Abstract

In this paper, we construct toric data of moduli space of quasi maps of degree $d$ from P^{1} with two marked points to weighted projective space P(1.1,1,3). With this result, we prove that the moduli space is a compact toric orbifold. We also determine its Chow ring. Moreover, we give a proof of the conjecture proposed by Jinzenji that a series of intersection numbers of the moduli spaces coincides with expansion coefficients of inverse function of -log(j(tau)).