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Noncommutative Singularity Theory

Gavin Brown, Michael Wemyss

Abstract

This is an expository article on the noncommutative singularity theory of power series in noncommuting variables, its motivation from deformation theory, and its applications to contractibility of curves and the classification of smooth 3-fold flops.

Noncommutative Singularity Theory

Abstract

This is an expository article on the noncommutative singularity theory of power series in noncommuting variables, its motivation from deformation theory, and its applications to contractibility of curves and the classification of smooth 3-fold flops.

Paper Structure

This paper contains 19 sections, 8 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Noncommutative Singularity Theory
  3. Classical singularity theory
  4. Noncommutative singularity theory
  5. The singularity theory toolkit
  6. Proposal
  7. Deformation Theory
  8. Deformation theory framework
  9. Commutative deformation theory
  10. Noncommutative deformation theory
  11. Deforming multiple objects
  12. The setting of curves
  13. Contractions
  14. Which deformation theory?
  15. Link to noncommutative singularity theory
  16. ...and 4 more sections

Key Result

Theorem 2.3

If $f$ is a simple singularity in the sense of def:simple, then up to relabelling variables $z_1,\hdots z_{d-2},x,y$, and up to isomorphism (analytic or formal changes of coordinates in the domain), $f$ is one of where $z^{2}=z_1^{2}+\hdots+ z_{d-2}^{2}$.

Theorems & Definitions (19)

  • Definition 2.2
  • Theorem 2.3: e.g. Yoshino
  • Definition 2.5
  • Example 2.6
  • Remark 2.8
  • Definition 2.9
  • Remark 2.10
  • Definition 2.11
  • Theorem 2.12: Splitting Lemma, DWZ
  • Example 2.13
  • ...and 9 more