Cremona transformations and derived equivalences of K3 surfaces

Abstract

We exhibit a Cremona transformation of ${\mathbf{P}}^4$ such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show that the difference of the two K3 surfaces annihilates the class of the affine line in the Grothendieck ring of varieties.

29 pages. Published in Compositio Mathematica

References in corpus (4)

• The Fano variety of lines and rationality problem for a cubic hypersurface
• Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics
• Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties
• The class of the affine line is a zero divisor in the Grothendieck ring: via $G_2$-Grassmannians

Cited by in corpus (10)

• Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties
• Intersections of two Grassmannians in ${\mathbf{P}}^9$
• Cycle class maps and birational invariants
• L-equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces
• An example of birationally inequivalent projective symplectic varieties which are D-equivalent and L-equivalent
• Motives of derived equivalent K3 surfaces
• Equivalence of K3 surfaces from Verra threefolds
• On cylindrical smooth rational Fano fourfolds
• Descent theory of simple sheaves on $C_1$-fields
• New rational cubic fourfolds arising from Cremona transformations