Related papers: Higher Nash blowups
In his previous paper, the author has defined a higher version of the Nash blowup and considered it a possible candidate for the one-step resolution. In this paper, we will introduce another higher version of the Nash blowup and prove that…
The higher Nash blowup of an algebraic variety replaces singular points with limits of certain spaces carrying higher order data associated to the variety at non-singular points. In the case of normal toric varieties we give a combinatorial…
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…
It is a long-standing question whether an arbitrary variety is desingularized by finitely many normalized Nash blow-ups. We consider this question in the case of a toric variety. We interpret the normalized Nash blow-up in polyhedral terms,…
The Nash blowing-up (or modification) of an algebraic variety $X$ is a canonical process that produces a proper, birational morphism $\pi : X' \to X$ of varieties. It is expected that the singularities of $X'$ will be better than those of…
The higher Nash blowup of an algebraic variety replaces singular points with limits of certain spaces carrying higher-order data associated to the variety at non-singular points. In this note we will define a higher-order Jacobian matrix…
We prove that the higher Nash blowup of a normal toric variety defined over a field of positive characteristic is an isomorphism if and only if it is non-singular. We also extend a result by R. Toh-Yama which shows that higher Nash blowups…
Any Lie algebroid $A$ admits a Nash-type blow-up $\mathrm{Nash}(A)$ that sits in a nice short exact sequence of Lie algebroids $0\rightarrow K\rightarrow \mathrm{Nash}(A)\rightarrow \mathcal{D}\rightarrow 0$ with $K$ a Lie algebra bundle…
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…
Approximation of real analytic functions by Nash functions is a classical topic in real geometry. In this paper, we focus on the Nash approximation of an analytic desingularization of a Nash function germ obtained by a sequence of…
We compute the Nash blow-up of a cominuscule Schubert variety. In particular, we show that the Nash blow-up is algebraically isomorphic to another Schubert variety of the same Lie type. As a consequence, we give a new characterization of…
We show that the Nash blowup of 2-generic determinantal varieties over fields of positive characteristic is non-singular. We prove this in two steps. Firstly, we explicitly describe the toric structure of such varieties. Secondly, we show…
We study the higher Nash blow-ups introduced by T. Yasuda and investigate the higher version of the classical Nobile's theorem. In particular, we give a characteristic free proof of the higher Nobile's theorem for the graded case. We also…
It is shown that, for any reduced algebraic variety in characteristic zero, one can resolve all but simple normal crossings (snc) singularities by a finite sequence of blowings-up with smooth centres which, at every step, avoids points…
The Nash multiplicity sequence was defined by M. Lejeune-Jalabert as a non-increasing sequence of integers attached to a germ of a curve inside a germ of a hypersurface. M. Hickel generalized this notion and described a sequence of blow ups…
We introduce conditions on cones of normal toric varieties under which the polyhedron defining the normalized Nash blowup does not depend on the characteristic of the base field. As a consequence, we deduce several results on the resolution…
We present a complete classification of normal toric surfaces that are resolved by a single normalized Nash blowup. Likewise, we obtain a complete classification of those resolved by a single Nash blowup. In both cases, the classification…
In this paper we describe the implementation that led to the counterexamples to the Nash blowup conjectures recently discovered by the authors. We also provide new examples of toric varieties with prescribed singularities that are not…
We construct a series of blowups $(\widetilde M_i,\pi_i)_{i\in \mathbb N_0}$ of a singular foliation by applying to the universal Lie $\infty$-algebroid of a singular foliation the so-called Nash modification. For $i=0$, we recover a blowup…
In this paper we show that iterating Nash blowups or normalized Nash blowups does not resolve the singularities of algebraic varieties of dimension four or higher over an algebraically closed field of arbitrary characteristic.