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Higher Nash blow-ups and Nobile theorem

Shravan Saoji

TL;DR

The paper extends Nobiles's theorem to higher Nash blow-ups in a graded setting, providing a characteristic-free proof for the higher case and a characteristic zero proof for the second order. It develops a framework based on the module of principal parts, arc constructions, and Noether normalization to relate higher and classical Nash blow-ups. The main contributions show that if the higher Nash blow-up is isomorphic to the original variety, then the variety must be nonsingular under the stated characteristic assumptions, with the $n=2$ case proven in char $0$. This work strengthens the link between singularity detection via Nash-type constructions and classical resolution techniques, offering tools applicable to graded and normal cases.

Abstract

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 give a proof for the 2nd order Nash blow-ups in characteristic zero.

Higher Nash blow-ups and Nobile theorem

TL;DR

The paper extends Nobiles's theorem to higher Nash blow-ups in a graded setting, providing a characteristic-free proof for the higher case and a characteristic zero proof for the second order. It develops a framework based on the module of principal parts, arc constructions, and Noether normalization to relate higher and classical Nash blow-ups. The main contributions show that if the higher Nash blow-up is isomorphic to the original variety, then the variety must be nonsingular under the stated characteristic assumptions, with the case proven in char . This work strengthens the link between singularity detection via Nash-type constructions and classical resolution techniques, offering tools applicable to graded and normal cases.

Abstract

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 give a proof for the 2nd order Nash blow-ups in characteristic zero.

Paper Structure

This paper contains 10 sections, 17 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reducing higher Nobile's theorem to classical Nobile's theorem
  4. Arc lemma
  5. Construction
  6. n property for the graded case
  7. 2 property for R
  8. Filtration and associated graded ring of PRn
  9. n property for G
  10. Higher Nobile's theorem

Key Result

Theorem A

Let $X$ be a graded algebraic variety over an algebraically closed field $k$. Suppose $\mathrm{Nash}_n(X) \cong X$ for some $n \in \mathbb{N}$, then

Theorems & Definitions (43)

  • Theorem A: see \ref{['Higher_Nash_graded']}
  • Theorem B: see \ref{['Higher_Nash_n=2']}
  • Theorem C: see \ref{['NashntoNashO']}
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • ...and 33 more