Cremona transformations and derived equivalences of K3 surfaces

TL;DR

This paper constructs a Cremona transformation in projective 4-space linking two derived equivalent, non-isomorphic K3 surfaces, and explores their implications in algebraic geometry.

Contribution

It provides an explicit example of a Cremona transformation connecting two derived equivalent K3 surfaces with distinct isomorphism classes.

Findings

The two K3 surfaces are derived equivalent but not isomorphic.

The difference of the two K3 surfaces affects the class of the affine line in the Grothendieck ring.

An explicit Cremona transformation in P^4 is constructed.

Abstract

We exhibit a Cremona transformation of ${\bf P}^4$ 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.