Surveys in Differential Geometry
Volume 14 (2009)
GIT constructions of moduli spaces of stable curves and maps
Pages: 315 – 370
Gieseker’s plan for using GIT to construct the moduli spaces of stable curves, now over 30 years old, has recently been extended to moduli spaces of pointed stable curves and of stable maps by Swinarski and Baldwin. The extensions turn out to be surprisingly delicate and both require the development of novel techniques for checking stability of Hilbert points. Simultaneously, interest in the area has been spurred by the log minimal model program of Hassett and his coworkers Hyeon and Lee in which these models are produced by suitably modified GIT constructions. Here I first give an introduction to the area by sketching Gieseker’s strategy. Then I review a number of variants—those involving unpointed curves that arise in Hassett’s program emphasizing Schubert’s moduli space of pseudostable curves, that of Swinarski for weighted pointed stable curves, and that of Baldwin and Swinarski for pointed stable maps—focusing on the steps at which new ideas are needed. Finally, I list open problems in the area, particularly some arising in the log minimal model program that seem inaccessible to current techniques.