Nash blowups in prime characteristic

Abstract

We initiate the study of Nash blowups in prime characteristic. First, we show that a normal variety is non-singular if and only if its Nash blowup is an isomorphism, extending a theorem by A. Nobile. We also study higher Nash blowups, as defined by T. Yasuda. Specifically, we give a characteristic-free proof of a higher version of Nobile's theorem for quotient varieties and hypersurfaces. We also prove a weaker version for F -pure varieties.Main Theorem (see Theorem 3.10). Let X be a normal irreducible variety. If Nash 1 .X / Š X, then X is a non-singular variety.

Factorization of normalization of Nash blow-up of order n of An by the minimal resolution

“…In particular, the Nash blowup try to solve singularities by iterating the process, while higher Nash looks for resolution of singularities in one step. Those questions have been treated in [12,14,7,8,11,17,19,9,10,1,4,5,18,6].…”

“…The strategy for this special case is translate the original geometric problem into a combinatorial one and then try to solve it in this context. One particular case of this is [18], where shows that the n-th Nash blowup of the toric surface singularity A 3 is singular for any n > 0 (This also happens in prime characteristic [6]). Moreover, the computational evidence of [18] points out that the same thing happens for any A n .…”