Partial desingularizations of good moduli spaces of Artin toric stacks
Let \(\mathbb{C}\) be the base field. Let \(G\) be a reductive linear algebraic group acting properly on a smooth projective variety \(X\). By Geometric Invariant Theory, one has open subsets \(X^s \subset X^{ss} \subset X\) of stable and semi-stable points. The quotient \(X^{ss}/G\) is projective and generally singular; and the quotient \(X^{s}/G\) has at most finite quotient singularities. \textit{F. C. Kirwan} [Ann. Math. (2) 122, 41--85 (1985; Zbl 0592.14011)] described a procedure of blowing \(X\) up along a sequence of smooth \(G\)-invariant subvarieties to obtain a variety \(X'\) with a \(G\)-action, such that every semistable point of \(X'\) is stable. Hence, the quotient variety \((X')^{ss}/G\) is projective with only finite quotient singularities, and there is an induced projective birational morphism \((X')^{ss}/G \to X^{ss}/G\) which is an isomorphism over the open set \(X^s/G\). One can therefore consider \((X')^{ss}/G\) as a partial desingularization of \(X^{ss}/G\). In this paper, the authors study a similar construction when \(\mathcal{X}\) is an Artin toric stack. To do this, they use \textit{Reichstein transformations}, which are certain birational transformations of Artin stacks with good moduli spaces. Let \(\mathcal{C} \subset \mathcal{X}\) be a closed substack. Then the Reichstein transformation of \(\mathcal{X}\) relative to \(\mathcal{C}\) is defined to be the complement of the strict transform of the saturation of \(\mathcal{C}\) relative to the quotient map \(q: \mathcal{X} \to M\) in the blow-up of \(\mathcal{X}\) along \(\mathcal{C}\). Theorem 4.7 states that the Reichstein transformation of a toric stack along a toric substack is another toric stack, which can moreover be described combinatorially in terms of the original toric stack. Applying this result repeatedly gives the result.
scientific article