• Corpus ID: 228063934

# Factorization centers in dimension two and the Grothendieck ring of varieties

@article{Lin2020FactorizationCI,
title={Factorization centers in dimension two and the Grothendieck ring of varieties},
author={Hsueh-Yung Lin and Evgeny Shinder and Susanna Zimmermann},
journal={arXiv: Algebraic Geometry},
year={2020}
}
• Published 9 December 2020
• Mathematics
• arXiv: Algebraic Geometry
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…
2 Citations

### Motivic invariants of birational maps

• Mathematics
• 2022
. We construct invariants of birational maps with values in the Kontsevich–Tschinkel group and in the truncated Grothendieck groups of varieties. These invariants are morphisms of groupoids and are

### Burnside groups and orbifold invariants of birational maps

• Mathematics
• 2022
. We construct new invariants of equivariant birational isomorphisms taking values in equivariant Burnside groups.

## References

SHOWING 1-10 OF 45 REFERENCES

### Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics

• Mathematics
• 2016
We discuss a conjecture saying that derived equivalence of smooth projective simply connected varieties implies that the difference of their classes in the Grothendieck ring of varieties is

### The class of a torus in the Grothendieck ring of varieties

We establish a formula for the classes of certain tori in the Grothendieck ring of varieties, expressing them in terms of the natural lambda-structure on the Grothendieck ring. More explicitly, we

### The Fano variety of lines and rationality problem for a cubic hypersurface

• Mathematics
• 2014
We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in

### The annihilator of the Lefschetz motive

In this paper we study a spectrum $K(\mathcal{V}_k)$ such that $\pi_0 K(\mathcal{V}_k)$ is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric

### Relations in the Cremona group over perfect fields

For perfect fields $k$ satisfying $[\bar k:k]>2$, we construct new normal subgroups of the plane Cremona group and provide an elementary proof of its non-simplicity, following the melody of the

### Derived Categories of Families of Sextic del Pezzo Surfaces

• A. Kuznetsov
• Mathematics
International Mathematics Research Notices
• 2019
We construct a natural semiorthogonal decomposition for the derived category of an arbitrary flat family of sextic del Pezzo surfaces with at worst du Val singularities. This decomposition has

### A note on motivic integration in mixed characteristic

• Mathematics
• 2009
We introduce a quotient of the Grothendieck ring of varieties by identifying classes of universally homeomorphic varieties. We show that the standard realization morphisms factor through this

### Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

• Mathematics
• 2015
We study the birational properties of geometrically rational surfaces from a derived categorical perspective. In particular, we give a criterion for the rationality of a del Pezzo surface S over an