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

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.

Exposition improved, main conjecture slightly updated

Cited by in corpus (7)

• Cremona transformations and derived equivalences of K3 surfaces
• Quadric bundles and hyperbolic equivalence
• L-equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces
• Equivalence of K3 surfaces from Verra threefolds
• Residual categories of quadric surface bundles
• Invariants of nested Hilbert and Quot schemes on surfaces
• Derived categories of quartic double fivefolds