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Nash blowups of toric varieties in prime characteristic

Daniel Duarte, +2 more
- 11 Aug 2022 - 
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TLDR
In this paper , it was shown that the iteration of normalized Nash blowups desingularizes normal toric surfaces and that the Nash blowup of a toric variety of arbitrary dimension in prime characteristic can be described combinatorially.
Abstract: 
Abstract We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime characteristic version of the logarithmic Jacobian ideal of a toric variety and prove that its blowup coincides with the Nash blowup of the variety. As a consequence, the Nash blowup of a, not necessarily normal, toric variety of arbitrary dimension in prime characteristic can be described combinatorially.

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References
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Gröbner bases and convex polytopes

Bernd Sturmfels
TL;DR: Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The second hypersimplex $\mathcal A$-graded algebras Canonical subalgebra bases Generators, Betti numbers and localizations Toric varieties in algebraic geometry as mentioned in this paper.
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Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties

Tadao Oda
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Günter Ewald
- 01 May 1985 - 
TL;DR: In this paper, a new area of research has developed relating two subjects which until now had very little in common: convexity and algebraic geometry, and a characterization of Milnor's number of critical points of complex algebraic functions by numbers assigned to convex polytopes is presented.
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Matroid Theory (Oxford Graduate Texts in Mathematics)

James G. Oxley
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Sandwiched singularities and desingularization of surfaces by normalized Nash transformations

Mark Spivakovsky
- 01 May 1990 - 
TL;DR: In this article, a succession of normalized Nash transformations are used to resolve the singularities of a complex surface, and a classification of sandwiched surface singularities is given for complex surfaces.