Generating the plane Cremona groups by involutions

Lamy, Stéphane; Schneider, Julia

doi:10.48550/arxiv.2110.02851

Cited by 2 publications

(2 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 3.7. A different proof of Theorem 3.5 in the case when the surfaces X and Y are rational can be deduced from a more recent result by Lamy and Schneider [LS21], who analyze relations between Sarkisov links to prove that Cr 2 (k) is generated by involutions. This implies that every homomorphism from Cr 2 (k) to a free abelian group is trivial (for k = R one can also deduce this from [Zim18,Theorem 1.1]).…”

Section: Formulation Of the Main Resultsmentioning

confidence: 99%

Factorization centers in dimension 2 and the Grothendieck ring of varieties

Lin,

Shinder,

Zimmermann

2023

Alg. Geom.

View full text Add to dashboard Cite

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 ϕ : X 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 ϕ. This implies that nontrivial L-equivalences of zero-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 zero-dimensional variety intrinsic to X, which is blown up in any rationality construction of X.

show abstract

Section: Formulation Of the Main Resultsmentioning

confidence: 99%

Factorization centers in dimension 2 and the Grothendieck ring of varieties

Lin,

Shinder,

Zimmermann

2023

Alg. Geom.

View full text Add to dashboard Cite

show abstract

“…In this article, we will focus on Cremona groups Bir(P 2 k ) over finite fields k -a subject that has attracted substantial interest recently, also because of its connections to cryptography (see for instance [SB21], [Sch20], [SZ20], [ALNZ19], [LS21]). One of the reasons why birational transformations of surfaces are well understood, is the fact that they can be factorized into a sequence of blow-ups of points and contractions of curves.…”

Section: Cremona Groupsmentioning

confidence: 99%

Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes

Genevois¹,

Lonjou²,

Urech³

2021

Preprint

View full text Add to dashboard Cite

We show that plane Cremona groups over finite fields embed into Neretin groups, i.e., groups of almost automorphisms of rooted trees. The image of this embedding is dense if our base field has odd characteristic or is equal to F2. This is no longer true, if the finite base field has even characteristic and contains at least 4 elements; for this case we show that the permutations induced by birational transformations on rational points of regular projective surfaces are always even. In a second part, we construct explicit locally compact CAT(0) cube complexes, on which Neretin groups act properly. This allows us to recover in a unified way various results on Neretin groups such as that they are of type F∞. We also prove a new fixed-point theorem for CAT(0) cube complexes without infinite cubes and use it to deduce a regularisation theorem for plane Cremona groups over finite fields.2010 Mathematics subject classification.

show abstract

Generating the plane Cremona groups by involutions

Abstract: We prove that over any perfect field the plane Cremona group is generated by involutions.Corollary 1.2. Let k be a perfect field. The abelianization of the Cremona group Bir k (P 2 ) is a group of exponent 2 (any non-trivial element has order 2).

Cited by 2 publications

References 6 publications

Factorization centers in dimension 2 and the Grothendieck ring of varieties

Factorization centers in dimension 2 and the Grothendieck ring of varieties

Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes

Contact Info

Product

Resources

About