# 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 $X$.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-03063890
Contributor : Susanna Zimmermann Connect in order to contact the contributor
Submitted on : Monday, December 14, 2020 - 11:59:55 AM
Last modification on : Friday, January 7, 2022 - 9:52:02 AM

### Identifiers

• HAL Id : hal-03063890, version 1
• ARXIV : 2012.04806

### Citation

Hsueh-Yung Lin, Evgeny Shinder, Susanna Zimmermann. Factorization centers in dimension two and the Grothendieck ring of varieties. 2020. ⟨hal-03063890⟩

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