Moduli space of stable maps to projective space via GIT

TL;DR

This paper analyzes the birational relationship between the moduli space of genus 0 stable maps and the quasi-map space to projective space for degree 3, explicitly describing the blow-up and blow-down processes and computing topological invariants.

Contribution

It explicitly describes the birational transformations connecting the moduli spaces for degree 3 and computes their topological invariants.

Findings

The birational map is a composition of three blow-ups and two blow-downs.

Centers of blow-up/down are explicitly identified in terms of lower degree moduli spaces.

Betti numbers, Picard group, and cohomology ring are calculated for the degree 3 case.

Abstract

We compare the Kontsevich moduli space of genus 0 stable maps to projective space with the quasi-map space when $d=3$. More precisely, we prove that when $d=3$, the obvious birational map from the quasi-map space to the moduli space of stable maps is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down centers explicitly in terms of the moduli spaces for lower degrees. Using this, we calculate the Betti numbers, the integral Picard group, and the rational cohomology ring. The degree two case is worked out as a warm-up.