Existence of moduli spaces for algebraic stacks

Abstract

We provide necessary and sufficient conditions for when an algebraic stack admits a good moduli space and prove a semistable reduction theorem for points of algebraic stacks equipped with a $\Theta$-stratification. These results provide a generalization of the Keel--Mori theorem to moduli problems whose objects have positive dimensional automorphism groups and give criteria on the moduli problem to have a separated or proper good moduli space. To illustrate our method, we apply these results to construct proper moduli spaces parameterizing semistable $\mathcal{G}$-bundles on curves and moduli spaces for objects in abelian categories.

79 pages: compatible with final published version

References in corpus (2)

• The étale local structure of algebraic stacks
• Fibration de Hitchin et endoscopie

Cited by in corpus (24)

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• Moduli of objects in finite length abelian categories
• Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity
• Moduli spaces of semistable modules over Lie algebroids
• On two families of Enriques categories over K3 surfaces