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 $X$.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2020
- DOI:
- 10.48550/arXiv.2012.04806
- arXiv:
- arXiv:2012.04806
- Bibcode:
- 2020arXiv201204806L
- Keywords:
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- Mathematics - Algebraic Geometry
- E-Print:
- Final version. To appear in Algebraic Geometry