Existence and properties of geometric quotients (Q2857295)

In this paper, the author studies a condition known as ``strong geometricity'' that ensures that geometric quotients of algebraic spaces are categorical.NEWLINENEWLINERecall that a groupoid (in the category of algebraic spaces) is defined by two algebraic spaces, \(R\) and \(X\); and two morphisms \(s,t:R \to X\), subject to a number of axioms. (The definition is given in full detail in Definition 1.1.) An equivariant morphism is defined as a morphism \(q:X \to Y\) such that \(q \circ s = q \circ t\). \(q\) is called a strongly geometric quotient if it is geometric, universally Zariski, universally constructible and the canonical morphism \(R \to X \times_Y X\) is universally submersive. In Theorem 3.16, the author proves that, a strongly geometric quotient \(q\) is categorical under any one of a list of conditions on \(q\). This generalizes Corollary 2.15 in [\textit{J. Kollár}, Ann. Math. (2) 145, No. 1, 33--79 (1997; Zbl 0881.14017)]. The author also gives results on the quotients of finite locally free groupoids of affine schemes in Section 4, and uses these to deduce existence of quotients of algebraic spaces by finite groups in Section 5. In Section 6, it is proved that algebraic stacks with finite inertia have coarse moduli spaces (Theorem 6.12). This generalizes a result of \textit{S. Keel} and \textit{S. Mori} [Ann. Math. (2) 145, No. 1, 193--213 (1997; Zbl 0881.14018)].