Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Flasque quasi-resolutions of algebraic varieties

Mattia Pirani

Abstract

Flasque resolutions play an important role in understanding birational properties of algebraic tori. For instance, Colliot-Thélène and Sansuc have used them to compute $R$-equivalence classes of algebraic tori. We extend this notion to a larger class of algebraic varieties, including homogeneous spaces. This leads to a lower bound on the number of $R$-equivalence classes of homogeneous spaces, which is a slightly stronger version of a theorem of Colliot-Thélène and Kunyavskii.

Flasque quasi-resolutions of algebraic varieties

Abstract

Flasque resolutions play an important role in understanding birational properties of algebraic tori. For instance, Colliot-Thélène and Sansuc have used them to compute -equivalence classes of algebraic tori. We extend this notion to a larger class of algebraic varieties, including homogeneous spaces. This leads to a lower bound on the number of -equivalence classes of homogeneous spaces, which is a slightly stronger version of a theorem of Colliot-Thélène and Kunyavskii.
Paper Structure (11 sections, 60 theorems, 8 equations)

This paper contains 11 sections, 60 theorems, 8 equations.

Table of Contents

  1. Flasque tori and flasque quasi-resolutions
  2. Definitions and general properties
  3. Existence and uniqueness of flasque quasi-resolutions
  4. Special case
  5. General case
  6. Evaluation map for homogeneous spaces
  7. R-equivalence on homogeneous spaces
  8. Second construction of flasque quasi-resolutions
  9. General properties of the evaluation map
  10. On a theorem of Colliot-Thélène and Kunyavskiı̆
  11. On algebraic groups and torsors

Key Result

Proposition 1

Let $X$ be a algebraic $k$-variety. Suppose that $X(k)$ is non-empty and fix a point $x\in X(k)$. There exists a flasque quasi-resolution of $(X,x)$ if and only if $\emph{Pic}(\overline{X})$ is finitely generated.

Theorems & Definitions (165)

  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Proposition 1.6
  • proof
  • ...and 155 more