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An algebro-geometric model for the configuration category

Pedro Boavida de Brito, Geoffroy Horel, Danica Kosanović

Abstract

Using log-geometry, we construct a model for the configuration category of a smooth algebraic variety. As an application, we prove the formality of certain configuration spaces.

An algebro-geometric model for the configuration category

Abstract

Using log-geometry, we construct a model for the configuration category of a smooth algebraic variety. As an application, we prove the formality of certain configuration spaces.

Paper Structure

This paper contains 25 sections, 25 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Categories of Forests
  3. Forests
  4. Forests with levels
  5. Blow-ups
  6. Complex and oriented real blow-ups
  7. Iterated real blow-ups
  8. Blowing-up a submanifold
  9. Blowing-up a subscheme
  10. Log schemes
  11. Log schemes and their blow-ups
  12. The Kato--Nakayama realization
  13. Compactifications of configuration spaces
  14. Stratified spaces
  15. The Axelrod--Singer compactification
  16. ...and 10 more sections

Key Result

Theorem 1

Let $X$ be a smooth algebraic variety over a field $K$. There is a functor from $\Delta\mathrm{Fin}^\mathrm{op}$ to log-schemes over $K$ -- denoted $\mathrm{con}^{log}(X)$ -- with the following properties.

Theorems & Definitions (73)

  • Theorem
  • Corollary
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Definition 3.1
  • Definition 3.2
  • ...and 63 more