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Stacky Resolutions of Singular Schemes

  • Author(s): Satriano, Matthew
  • Advisor(s): Olsson, Martin C
  • et al.
Abstract

Given a singular scheme X over a field k, we consider the problem of resolving the singularities of X by an algebraic stack. When X is a toroidal embedding or is etale locally the quotient of a smooth scheme by a linearly reductive group scheme, we show that such &ldquo stacky resolutions &rdquo exist. Moreover, these resolutions are canonical and easily understandable in terms of the singularities of X.

We give three applications of our stacky resolution theorems: various generalizations of the Chevalley-Shephard-Todd Theorem, a Hodge decomposition in characteristic p, and a theory of toric Artin stacks extending the work of Borisov-Chen-Smith. While these applications are seemingly different, they are all related by the common theme of using stacky resolutions to study singular schemes.

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