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Desingularization of double covers of regular surfaces

Qing Liu

Abstract

Let $Z$ be a noetherian integral excellent regular scheme of dimension 2. Let $Y$ be an integral normal scheme endowed with a finite flat morphism $Y \to Z$ of degree 2. We give a description of Lipman's desingularization of $Y$ by explicit equations, leading to a desingularization algorithm for $Y$.

Desingularization of double covers of regular surfaces

Abstract

Let be a noetherian integral excellent regular scheme of dimension 2. Let be an integral normal scheme endowed with a finite flat morphism of degree 2. We give a description of Lipman's desingularization of by explicit equations, leading to a desingularization algorithm for .

Paper Structure

This paper contains 21 sections, 23 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. The multiplicity lambdapY
  3. Definition and basic properties for local rings
  4. Algorithm for determining the multiplicity lambdap(Y)
  5. Multiplicity in the global case
  6. Normalized blowing-up
  7. Blowing-up regular schemes along a point
  8. Normalization after blowing-up
  9. Lipman's resolution of singularities
  10. The exceptional locus of Lipman's resolution of singularities
  11. A complete example of resolution of singularities
  12. Simultaneous resolution of singularities
  13. Variations of the multiplicities after normalized blowing-ups
  14. Double covers of PP1k
  15. Multiplicity along a regular closed subscheme
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $Y$ be a noetherian integral normal excellent surface. Consider a sequence of birational morphisms where each $Y_{i+1} \to Y_i$ is the normalization of the blowing-up of $Y_i$ along some singular closed (reduced) point. Then the sequence is necessarily finite.

Theorems & Definitions (57)

  • Theorem 1.1: Lipman, Lip
  • definition nameusethethm
  • definition nameusethethm
  • Lemma 2.1
  • proof
  • definition nameusethethm
  • remark nameusethethm
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 47 more