Flasque quasi-resolutions of algebraic varieties

TL;DR

This paper extends the concept of flasque resolutions from algebraic tori to a broader class of algebraic varieties, including homogeneous spaces, providing new insights into their birational properties and $R$-equivalence classes.

Contribution

It introduces a generalized notion of flasque resolutions applicable to more varieties, leading to improved bounds on $R$-equivalence classes of homogeneous spaces.

Findings

Established a lower bound on the number of $R$-equivalence classes for homogeneous spaces.

Extended the use of flasque resolutions beyond algebraic tori.

Provided a stronger version of existing theorems on $R$-equivalence.

Abstract

Flasque resolutions play an important role in understanding birational properties of algebraic tori. For instance, Colliot-Th\'{e}l\`{e}ne 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-Th\'el\`ene and Kunyavskii.