Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics

TL;DR

This paper explores a conjecture linking derived equivalence of certain varieties to their classes in the Grothendieck ring, providing evidence through examples and a new case involving quadrics and double covers.

Contribution

It offers new support for the conjecture by analyzing a specific family of quadrics and their associated double covers, demonstrating the conjecture's validity in this context.

Findings

Derived equivalence implies the difference in classes is annihilated by the affine line class.

Supports the conjecture with known examples and a new case involving quadrics.

Shows the vanishing of the Brauer class leads to the conjecture's condition.

Abstract

We discuss a conjecture saying that derived equivalence of simply connected smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection $X$ of three quadrics in ${\mathbf P}^5$ and the corresponding double cover $Y \to {\mathbf P}^2$ branched over a sextic curve. We show that as soon as the natural Brauer class on $Y$ vanishes, so that $X$ and $Y$ are derived equivalent, the difference $[X] - [Y]$ is annihilated by the affine line class.