Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characteristic-free normalized Nash blowup of toric varieties

Federico Castillo, Daniel Duarte, Maximiliano Leyton-Álvarez, Alvaro Liendo

TL;DR

This work identifies when the combinatorics of the normalized Nash blowup for toric varieties are independent of the base field's characteristic. By introducing $G$-stability and $G$-flatness for cones and lattice polytopes, the authors prove that $\mathcal{N}_0(\sigma)=\mathcal{N}_p(\sigma)$ for all primes $p$ whenever $\sigma^{\vee}$ is $G$-stable, enabling characteristic-free conclusions about resolution properties. They establish a $G$-desingularization result via refined, $G$-stable subdivisions and connect these to the normalized Nash blowup through the polyhedron $\mathcal{N}_p(\sigma)$, recovering known positive-characteristic results and constructing new families with non-singular normalized Nash blowups in arbitrary characteristic using the barycentric hull of smooth, $G$-flat polytopes. The barycentric hull framework provides a robust mechanism to deduce regularity of feasible cones and one-step resolutions in dimension three contexts, highlighting the interplay between combinatorics of lattice polytopes and geometric resolution properties in toric settings.

Abstract

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 of singularities properties of normalized Nash blowups. In particular, we recover all known results of the families that can be resolved via normalized Nash blowups in positive characteristic. We also provide new families of toric varieties whose normalized Nash blowup is non-singular in arbitrary characteristic.

Characteristic-free normalized Nash blowup of toric varieties

TL;DR

This work identifies when the combinatorics of the normalized Nash blowup for toric varieties are independent of the base field's characteristic. By introducing -stability and -flatness for cones and lattice polytopes, the authors prove that for all primes whenever is -stable, enabling characteristic-free conclusions about resolution properties. They establish a -desingularization result via refined, -stable subdivisions and connect these to the normalized Nash blowup through the polyhedron , recovering known positive-characteristic results and constructing new families with non-singular normalized Nash blowups in arbitrary characteristic using the barycentric hull of smooth, -flat polytopes. The barycentric hull framework provides a robust mechanism to deduce regularity of feasible cones and one-step resolutions in dimension three contexts, highlighting the interplay between combinatorics of lattice polytopes and geometric resolution properties in toric settings.

Abstract

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 of singularities properties of normalized Nash blowups. In particular, we recover all known results of the families that can be resolved via normalized Nash blowups in positive characteristic. We also provide new families of toric varieties whose normalized Nash blowup is non-singular in arbitrary characteristic.
Paper Structure (8 sections, 21 theorems, 54 equations, 3 figures)

This paper contains 8 sections, 21 theorems, 54 equations, 3 figures.

Table of Contents

  1. Preliminaries on toric varieties
  2. G-stability
  3. G-desingularization
  4. The normalized Nash blowup of a normal toric variety
  5. Known results on normalized Nash blowups of toric varieties
  6. G-flat Hilbert bases and one-step resolution
  7. Barycentric hull
  8. One-step resolution via normalized Nash blowups in arbitrary characteristic

Key Result

Theorem 1

(see th:Nash polyh and cor:char_free) Let $\sigma$ be a full-dimensional strongly convex cone. Suppose that $\sigma^\vee$ is $G$-stable or it is defined by a smooth $G$-flat lattice polytope. Then $\mathcal{N}_0(\sigma)=\mathcal{N}_p(\sigma)$ for all $p>0$ prime.

Figures (3)

  • Figure 1: Barycentric hull of the rhombus (the points in $P\cap M$ are dots and the barycenters are crosses, the origin is not a barycenter).
  • Figure 2: Barycentric hull of a smooth polygon (the points in $P\cap M$ are dots and the barycenters are crosses, the origin is a barycenter).
  • Figure 3: A polygon with its corners highlighted.

Theorems & Definitions (61)

  • Theorem
  • Theorem
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 51 more