A geometric invariant theory construction of moduli spaces of stable maps

TL;DR

This paper constructs moduli spaces of stable maps using geometric invariant theory, providing a new GIT-based approach that extends to certain classical moduli spaces over the integers.

Contribution

It introduces a novel GIT construction for moduli spaces of stable maps, including a new proof for the nonemptiness of the semistable set.

Findings

GIT construction of ar M_g,n(P^r,d) over Spec C

Special case GIT presentation of ar M_g,n over Spec Z

Different proof for nonemptiness of semistable set

Abstract

We construct the moduli spaces of stable maps, \bar M_g,n(P^r,d), via geometric invariant theory (GIT). This construction is only valid over Spec C, but a special case is a GIT presentation of the moduli space of stable curves of genus g with n marked points, \bar M_g,n; this is valid over Spec Z. Our method follows that used in the case n=0 by Gieseker to construct \bar M_g, though our proof that the semistable set is nonempty is entirely different.