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Desingularization of binomial varieties using toric Artin stacks

Dan Abramovich, Bernd Schober

Abstract

We show how the notion of fantastacks can be used to effectively desingularize binomial varieties defined over algebraically closed fields. In contrast to a desingularization via blow-ups in smooth centers, we drastically reduce the number of steps and the number of charts appearing along the process. Furthermore, we discuss how our considerations extend to a partial simultaneous normal crossings desingularization of finitely many binomial hypersurfaces.

Desingularization of binomial varieties using toric Artin stacks

Abstract

We show how the notion of fantastacks can be used to effectively desingularize binomial varieties defined over algebraically closed fields. In contrast to a desingularization via blow-ups in smooth centers, we drastically reduce the number of steps and the number of charts appearing along the process. Furthermore, we discuss how our considerations extend to a partial simultaneous normal crossings desingularization of finitely many binomial hypersurfaces.
Paper Structure (24 sections, 9 theorems, 34 equations)

This paper contains 24 sections, 9 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Key ingredient: fantastacks
  3. Setup: binomials in toric varieties
  4. Arrangements of binomials and binomial subschemes
  5. Simple arrangements and problematic primes
  6. Subdivisions and modifications associated to binomials
  7. Main result on arrangements of binomials
  8. Remarks
  9. Main result on binomial varieties
  10. Prior work
  11. Computational motivation
  12. Summary
  13. Acknowledgments
  14. Basics on fantastacks
  15. Binomial hypersurfaces: generalities
  16. ...and 9 more sections

Key Result

Theorem 1

Let $K$ be an algebraically closed field of arbitrary characteristic $p \geq 0$. Let $f_1, \ldots, f_a \in K[x] = K [x_1, \ldots, x_m]$ be finitely many binomials, where $a, m \in \mathbb{Z}_+$ with $m \geq 2$. Let $\mathcal{E} \subset \mathbb{Z}$ be the set of problematic primes associated to the

Theorems & Definitions (25)

  • Theorem 1: See Theorem \ref{['Thm:1Text']}
  • Theorem 2: See Theorem \ref{['Thm:2Text']}
  • Definition 2.1: Toric1
  • Definition 2.2: Toric1
  • Definition 2.3: Toric1
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 4.1
  • Proposition 4.2
  • ...and 15 more