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 Lequivalences of $0$dimensional varieties cannot be constructed based on birational automorphisms of a surface. It also implies that rationality centers are welldefined for every rational surface $X$, namely there exists a $0$dimensional variety intrinsic to $X$, which is blown up in any rationality construction of $X$.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.04806
 Bibcode:
 2020arXiv201204806L
 Keywords:

 Mathematics  Algebraic Geometry