Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher Nash blowups

Takehiko Yasuda

Abstract

For each non-negative integer $n$, we define the $n$-th Nash blowup of an algebraic variety, and call them all higher Nash blowups. When $n=1$, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its $n$-th Nash blowup with $n$ large enough.

Higher Nash blowups

Abstract

For each non-negative integer , we define the -th Nash blowup of an algebraic variety, and call them all higher Nash blowups. When , it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its -th Nash blowup with large enough.

Paper Structure

This paper contains 16 sections, 22 theorems, 84 equations.

Table of Contents

  1. Definition and several constructions
  2. Definition
  3. $\psi_{n}$ is an isomorphism in char. 0
  4. Construction with the relative Hilbert scheme
  5. The Nash blowup associated to a coherent sheaf
  6. Formal completion
  7. General properties
  8. Compatibility with etale morphisms
  9. Group actions
  10. Conductor and Jacobian ideals
  11. Separation of analytic branches
  12. Higher Nash blowups of curves
  13. A deformation-theoretic criterion for the regularity
  14. Associated numerical monoids
  15. Conjecture \ref{['Yasuda-conjecture']} for curves
  16. ...and 1 more sections

Key Result

Proposition 2

Let $[Z] \in \mathbf{Nash}_{n} (X)$ with $Z \nsubseteq C$. Then $Z$ is contained in a unique analytic branch of $X$.

Theorems & Definitions (53)

  • Conjecture 1
  • Proposition 2: =Proposition \ref{['prop-general-separation']}
  • Theorem 3: =Theorem \ref{['thm-curve1']}
  • Corollary 4: =Corollary \ref{['cor-general-curve']}
  • Remark 5
  • Lemma 1.1
  • proof
  • Definition 1.2
  • Proposition 1.3
  • proof
  • ...and 43 more