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Article

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

Selecta Mathematica, New Series. 2018. Vol. 24. No. 4. P. 3475-3500.
Kuznetsov A. G., Shinder E.

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 P5 and the corresponding double cover Y→P2 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.