Factorization centers in dimension two and the Grothendieck ring of varieties

Abstract

We initiate the study of factorization centers of birational maps, and complete it for surfaces over a perfect field in this article. We prove that for every birational automorphism φ : X X of a smooth projective surface X over a perfect field k, the blowup centers are isomorphic to the blowdown centers in every weak factorization of φ. This implies that nontrivial L-equivalences of 0-dimensional varieties cannot be constructed based on birational automorphisms of a surface. It also implies that rationality centers are well-defined for every rational surface X, namely there exists a 0-dimensional variety intrinsic to X, which is blown up in any rationality construction of X.

Involutions on K3 surfaces and derived equivalence

“…The difference of classes of exceptional loci in (9.1) is nonzero due to Proposition 26 below. This gives an instance where the refinement of the invariant c(φ) in [LSZ20], [LS22] using group actions yields new information.…”