[Paper Review] Functorial resolution of tame quotient singularities in positive characteristic

Federico Buonerba|arXiv (Cornell University)|Nov 2, 2015

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes a functorial resolution of tame quotient singularities in positive characteristic for quasi-projective varieties over algebraically closed fields. Using an invariant-based inductive approach with weighted blow-ups along smooth centers and leveraging étale-local functoriality, the authors reduce the problem to tame cyclic quotient singularities and then further decrease the invariant via a global character condition, ultimately constructing a smooth, quasi-projective resolution that commutes with étale base change.

ABSTRACT

The object of the present is a proof of the existence of functorial resolution of tame quotient singularities for quasi-projective varieties over algebraically closed fields.

Motivation & Objective

• To prove the existence of a functorial resolution for quasi-projective varieties with tame quotient singularities in positive characteristic.
• To overcome the local-to-global obstruction in resolution by ensuring compatibility with étale base change.
• To reduce the resolution problem to tame cyclic quotient singularities using a global invariant and a character condition.
• To extend resolution techniques from toric geometry to positive characteristic by weakening toroidal assumptions via a faithful character condition.
• To demonstrate that the method fails under non-tame assumptions, suggesting a need for fundamentally different strategies in the general case.

Proposed method

• Define an invariant $ i(X) \in \mathbb{N} $ such that $ i(X) = 0 $ if and only if $ X $ is smooth, and proceed by induction on $ i(X) $.
• Use a two-step process: first reduce general tame quotient singularities to tame cyclic quotient singularities via birational modifications that preserve the invariant.
• Introduce a global character condition on a divisor $ E $ such that local cyclic stabilizers act faithfully across $ E $, enabling the construction of a globally well-defined sheaf of algebras.
• Apply weighted blow-ups along smooth centers using the sheaf of algebras to produce a new variety with smaller stabilizers, even if non-tame singularities temporarily appear.
• Handle non-tame cyclic singularities via an iterative blow-up process based on the maximal non-tame stabilizer, using the faithfulness of actions on a divisor to ensure descent to tame singularities.
• Ensure functoriality under étale base change by constructing the resolution via an iterative, étale-local procedure that commutes with pullbacks.

Experimental results

Research questions

• RQ1 Can a functorial resolution of tame quotient singularities be constructed in positive characteristic for quasi-projective varieties?
• RQ2 How can the local-to-global problem in resolution be overcome without assuming toroidal embeddings or global torus actions?
• RQ3 What role does a globally defined character for local stabilizers play in enabling a global resolution procedure?
• RQ4 Can weighted blow-ups be used effectively in positive characteristic even when they produce non-tame singularities, provided a faithful character condition is satisfied?
• RQ5 Why does the method fail under non-tame group actions, and what does this imply for a general resolution of quotient singularities?

Key findings

• The main theorem establishes the existence of a resolution functor $ X \to (M(X), r_X) $, where $ M(X) $ is smooth, quasi-projective, and $ r_X: M(X) \to X $ is a proper, birational, relatively projective morphism that is an isomorphism over the smooth locus of $ X $.
• The resolution functor commutes with étale base change, ensuring compatibility with algebraic spaces, Deligne-Mumford stacks, and analytic spaces.
• The first step reduces any tame quotient singularity to a tame cyclic quotient singularity while preserving the invariant, using étale-local functoriality.
• The second step reduces the invariant by constructing a weighted blow-up along a smooth center defined by a sheaf of algebras induced by faithful characters on a divisor, even when non-tame singularities arise temporarily.
• A key lemma (II.II.1) proves that if non-tame diagonal cyclic singularities appear, they can be resolved via an iterative blow-up process based on the maximal non-tame stabilizer, provided the stabilizers are faithful on a divisor.
• The method fails for non-tame actions due to the absence of regular parameters that diagonalize the group action, suggesting a need for a fundamentally different strategy in the general case.