Let X be an Artin stack with good moduli space X → M . We define the Reichstein transform of X relative to a closed substack C ⊂ X to be the complement of the strict transform of the saturation of C in the blowup of X along C. The main technical result of the paper is that the Reichstein transform of a toric Artin stack relative to a toric substack is again a toric stack. Precisely, the Reichstein transform relative to a cone in a stacky fan is the toric stack determined by stacky star subdivision. This leads to our main theorem which states that for toric Artin stacks there is a canonical sequence of Reichstein transforms that produces a toric Deligne-Mumford stack. When the good moduli space of the toric stack is a projective toric variety our procedure can be interpreted in terms of Kirwan's [Kir] partial desingularization of geometric invariant theory quotients.

Let X be a nonsingular variety (with dim X 2) over an algebraically closed field k of characteristic zero. Let α : Spec kJtK → X be an arc on X, and let v = ord α be the valuation given by the order of vanishing along α. We describe the maximal irreducible sub-both algebraically, in terms of the sequence of valuation ideals of v, and geometrically, in terms of the sequence of infinitely near points associated to v. As a corollary, we get that v is determined by its sequence of centers. Also, when X is a surface, our construction also applies to any divisorial valuation v, and in this case C (v) coincides with the one introduced in [L. Ein, R. Lazarsfeld, M. Musta¸tǎ, Contact loci in arc spaces,

We investigate properties and describe examples of tilt-stable objects on a smooth complex projective threefold. We give a structure theorem on slope semistable sheaves of vanishing discriminant, and describe certain Chern classes for which every slope semistable sheaf yields a Bridgeland semistable object of maximal phase. Then, we study tilt stability as the polarisation ω gets large, and give sufficient conditions for tilt-stability of sheaves of the following two forms: 1) twists of ideal sheaves or 2) torsion-free sheaves whose first Chern class is twice a minimum possible value.

No abstract

There is a well-developed intersection theory on smooth Artin stacks with quasi-affine diagonal [Gil, Vis, EG98, Kre]. However, for Artin stacks whose diagonal is not quasi-finite the notion of the degree of a Chow cycle is not defined. In this paper we propose a definition for the degree of a cycle on Artin toric stacks whose underlying toric varieties are complete. As an application we define the Euler characteristic of an Artin toric stack with complete good moduli space -extending the definition of the orbifold Euler characteristic. An explicit combinatorial formula is given for 3-dimensional Artin toric stacks.