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 $\phi : X \dashrightarrow X$ 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 $\phi$. 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 $X$.