Factorization centers in dimension 2 and the Grothendieck ring of varieties

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 zero-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 zero-dimensional variety intrinsic to X, which is blown up in any rationality construction of X.