Good Moduli Spaces for Artin Stacks

By Jarod Alper. 2008

401 citations recorded by the source corpus.

We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford’s geometric invariant theory and tame stacks.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/640f1268bc276458c225bf2492cca4ae5c52496f

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Citations in the corpus

Adequate moduli spaces and geometrically reductive group schemes (2010, neutral): “riterion: if f: X→Y is a quasi-compact and quasi-separated morphism of algebraic spaces, then f is adequately aﬃne if and only if f is aﬃne (see Theorem 4.3.1). The notion of a good moduli space (see [Alp08]) is deﬁned by replacing condition (1) with the requirement that the push-forward functor φ∗ ”
Variation of geometric invariant theory quotients and derived categories (2012, background): “art of the proposition, see also Corollary 2.2 of [Edi00]. Corollary 2.2 of loc. cit. also shows that X//LGis an algebraic space under the assumption that all stabilizers are trivial. Theorem 13.6 of [Alp08] shows that Y is a good moduli space of X//LGin general. Then, Theorem 6.6 of loc. cit. says ”
Uniqueness of K-polystable degenerations of Fano varieties (2018, background): “o varieties with volume V, satisfying Koll´ar’s condition is represented by an Artin stack MKss n,V of ﬁnite type and admits a projective good moduli space MKss n,V → M Kps n,V (in the sense of [Alp13]), whose closed points are in bijection with n-dimensional K-polystable Q-Fano varieties of volu”
Alternate Compactifications of Moduli Spaces of Curves (2010, background): “ ==SL(N+ 1) It only remains to check that ˚is a good moduli map. The rst property of good moduli maps, that they are categorical with respect to algebraic spaces, is a general feature of GIT [MFK94], [Alp08]. The second property, namely that ˚ 1(x) contains a unique closed point for each x2Hilbss;m ”
Mapping stacks and categorical notions of properness (2014, background): “ then a good quotient [S1] is a G-invariant map to an algebraic space q : X →Y such that q∗ : QC(X/G) →QC(Y) is t-exact, and q∗(OX)G = OY . This is a special case of the notion of a good moduli space [A1]. Example 4.2.4. Let X be a projective variety over a ﬁeld of characteristic 0. Let G be a reduc”
The étale local structure of algebraic stacks (2019, neutral): “ BSGL(E) ∼= BSGLn, and the composition X= [SpecA/G] →BSG→BSGL(E) ∼=BSGLn is aﬃne, that is X ∼=[SpecB/GLn,S]. By [Gro17, Thm. 1.1], X has the resolution property. Remark 2.6 (Aﬃneness). In contrast to [Alp13], we have only deﬁned linear reductivity for aﬃne group schemes G→S. We will however prove th”