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Functorial embedded resolution via weighted blowings up

Dan Abramovich, Michael Temkin, Jarosław Włodarczyk

Abstract

We provide a procedure for resolving, in characteristic 0, singularities of a variety $X$ embedded in a smooth variety $Y$ by repeatedly blowing up the worst singularities, in the sense of stack-theoretic weighted blowings up. No history, no exceptional divisors, and no logarithmic structures are necessary to carry this out; the steps are explicit geometric operations requiring no choices; and the resulting algorithm is efficient. A similar result was discovered independently by McQuillan in a manuscript posted on this archive simultaneously.

Functorial embedded resolution via weighted blowings up

Abstract

We provide a procedure for resolving, in characteristic 0, singularities of a variety embedded in a smooth variety by repeatedly blowing up the worst singularities, in the sense of stack-theoretic weighted blowings up. No history, no exceptional divisors, and no logarithmic structures are necessary to carry this out; the steps are explicit geometric operations requiring no choices; and the resulting algorithm is efficient. A similar result was discovered independently by McQuillan in a manuscript posted on this archive simultaneously.

Paper Structure

This paper contains 48 sections, 17 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Statement of result
  3. Our quest for the dream algorithm
  4. Weighted blowings up, stacks, and resolutions
  5. Invariants and parameters
  6. Tools and methods
  7. Concurrent and future work
  8. Examples: comparing smooth and weighted blowings up
  9. Blowing up without weights
  10. Weighted blowing up
  11. Efficiency
  12. The need for more centers
  13. Acknowledgements
  14. Valuative ideals, fractional ideals, and ${\mathbb{Q}}$-ideals
  15. Zariski--Riemann spaces
  16. ...and 33 more sections

Key Result

Theorem 1.1.1

There is a functor $F_{er}$, on pairs with smooth surjective morphisms, associating to a pair $X\subset Y$as above over a field in characteristic 0, with $X$singular, a center $\bar{J}$ with weighted blowing up $Y'\to Y$ and proper transform $F_{er}(X\subset Y) = (X' \subset Y')$, such that $Y'$ is

Theorems & Definitions (30)

  • Theorem 1.1.1
  • Remark 2.1.1
  • Lemma 3.4.1
  • Definition 4.1.1
  • Theorem 4.3.1
  • Proposition 4.4.1
  • proof
  • Theorem 5.1.1: ATW-principalization
  • proof
  • Corollary 5.1.2
  • ...and 20 more