Jacques Tits motivic measure

TL;DR

This paper introduces the Jacques Tits motivic measure, a new tool in algebraic geometry, to distinguish and relate various algebraic varieties via their classes in the Grothendieck ring, with applications to Severi-Brauer and quadric varieties.

Contribution

The paper constructs the Jacques Tits motivic measure and demonstrates its effectiveness in classifying algebraic varieties up to isomorphism or birational equivalence.

Findings

Two Severi-Brauer varieties associated to 2-torsion central simple algebras are isomorphic iff they have the same class.

Varieties associated to certain central simple algebras with periods 3-6 are birational if their classes coincide.

Quadric hypersurfaces are isomorphic iff they share the same class in the Grothendieck ring.

Abstract

In this article we construct a new motivic measure called the ${\it Jacques}$ ${\it Tits}$ ${\it motivic}$ ${\it measure}$. As a first main application of the Jacques Tits motivic measure, we prove that two Severi-Brauer varieties (or, more generally, two twisted Grassmannian varieties), associated to $2$-torsion central simple algebras, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that if two Severi-Brauer varieties, associated to central simple algebras of period $\{3, 4, 5, 6\}$, have the same class in the Grothendieck ring of varieties, then they are necessarily birational to each other. As a second main application of the Jacques Tits motivic measure, we prove that two quadric hypersurfaces (or, more generally, two involution varieties), associated to quadratic forms of dimension $6$ or to quadratic forms of arbitrary dimension defined over a base field $k$ with $I^3(k)=0$, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that the latter main application also holds for products of quadric hypersurfaces.