  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 $X$.
