GIT Constructions of Moduli Spaces of Stable Curves and Maps

Abstract

This largely expository paper first gives an introduction to Hilbert stability and its use in Gieseker's GIT construction of ${\overline{M}}_g$. Then I review recent work in this area--variants for unpointed curves that arise in Hassett's log minimal model program, starting with Schubert's moduli space of pseudostable curves, and constructions for weighted pointed stable curves and for pointed stable maps due to Swinarski and to Baldwin and Swinarski respectively. The focus is 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.

46 pages, 3 figures, written for Surveys in Differential Geometry

References in corpus (14)

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• The connectedness of the moduli space of maps to homogeneous spaces
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Cited by in corpus (4)

• Alternate Compactifications of Moduli Spaces of Curves
• Hilbert-Mumford criterion for nodal curves
• Groebner techniques for low degree Hilbert stability
• Stability of Tails and 4-Canonical Models