Universal flattening of Frobenius

Abstract

For a variety $X$ of positive characteristic and a non-negative integer $e$, we define its $e$-th F-blowup to be the universal flattening of the $e$-iterated Frobenius of $X$. Thus we have the sequence (a set labeled by non-negative integers) of blowups of $X$. Under some condition, the sequence stabilizes and leads to a nice (for instance, minimal or crepant) resolution. For tame quotient singularities, the sequence leads to the $G$-Hilbert scheme.

25 pages; v4. a full revision, notations changed, the isomorphism of the F-blowup and the G-Hilbert scheme has been generalized to the non-abelian case, errors corrected, the introduction shortened. v5. minor revision, to appear in the American Journal of Mathematics

References in corpus (4)

• Higher Nash blowups
• Noncommutative resolution, F-blowups and D-modules
• On monotonicity of F-blowup sequences
• Flag higher Nash blowups

Cited by in corpus (6)

• Noncommutative resolution, F-blowups and D-modules
• On monotonicity of F-blowup sequences
• F-blowups of normal surface singularities
• A higher-order tangent map and a conjecture on the higher Nash blowup of curves
• Frobenius morphisms of noncommutative blowups
• Non-commutative resolutions of linearly reductive quotient singularities