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Flag higher Nash blowups

Takehiko Yasuda

Abstract

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 it is compatible with products and smooth morphisms. We will also prove that the product of curves can be desingularized via both versions.

Flag higher Nash blowups

Abstract

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 it is compatible with products and smooth morphisms. We will also prove that the product of curves can be desingularized via both versions.

Paper Structure

This paper contains 9 sections, 13 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Convention
  3. Simple and flag higher Nash blowups
  4. Compatibility with products
  5. Compatibility with smooth morphisms
  6. Curves and products of curves
  7. Curves
  8. Various blowups of $\mathop{\mathrm{Spec}}\nolimits k[[x^{5},y^{7}]]$
  9. Product of curves

Key Result

Theorem 1.1

Let $X$ and $Y$ be varieties. Then there is a canonical isomorphism,

Theorems & Definitions (31)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.5
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • ...and 21 more