Recent zbMATH articles in MSC 14E07https://www.zbmath.org/atom/cc/14E072021-04-16T16:22:00+00:00WerkzeugSignature morphisms from the Cremona group over a non-closed field.https://www.zbmath.org/1456.140192021-04-16T16:22:00+00:00"Lamy, Stéphane"https://www.zbmath.org/authors/?q=ai:lamy.stephane"Zimmermann, Susanna"https://www.zbmath.org/authors/?q=ai:zimmermann.susannaThe Cremona group \(\operatorname{Cr}_2(\Bbbk)\) is the group of birational transformations of the projective plane \(\mathbb{P}^2\) over a field \(\Bbbk\). It was a long-standing question whether \(\operatorname{Cr}_n(\mathbb{C})\) is simple group. Several years ago \textit{S. Cantat} et al. made a breakthrough in [Acta Math. 210, No. 1, 31--94 (2013; Zbl 1278.14017)] by proving that \(\operatorname{Cr}_2(\mathbb{C})\) is not simple. \textit{A. Lonjou} generalized the result to an arbitrary field in [Ann. Inst. Fourier 66, No. 5, 2021--2046 (2016; Zbl 1365.14017)]. Over the field of complex numbers it was classically known that \(\operatorname{Cr}_2(\Bbbk)\) does not admit any non trivial homomorphism to an abelian group. Over the field of real numbers \textit{S. Zimmermann} proved in [Duke Math. J. 167, No. 2, 211--267 (2018; Zbl 1402.14015) ] that the abelianization of \(\operatorname{Cr}_2(\mathbb{R})\) is a direct sum of uncountably many \(\mathbb{Z}/2\mathbb{Z}\).
The article under review deals with an arbitrary perfect field with at least one Galois extension of degree eight. The authors constructed a tree on which \(\operatorname{Cr}_2(\Bbbk)\) acts so that \(\operatorname{Cr}_2(\Bbbk)\) can be written as an amalgam product by Bass-Serre theory. Note that each factor in the amalgam product is a big group and there are a lot of factors (same cardinality as the field \(\Bbbk\)). Consequently the authors constructed a homomorphism from \(\operatorname{Cr}_2(\Bbbk)\) to a free product of \(\mathbb{Z}/2\mathbb{Z}\), thus also a homomorphism from \(\operatorname{Cr}_2(\Bbbk)\) to a direct sum of \(\mathbb{Z}/2\mathbb{Z}\).
The tree mentionned above comes from a square complex constructed in this paper on which \(\operatorname{Cr}_2(\Bbbk)\) acts. The vertices of the square are rank \(r\) fibrations with \(r=1,2,3\); rank \(r\) fibrations are generalizations of Mori fiber spaces. Roughly speaking the edges and the faces of the square complex record Sarkisov links and relations among Sarkisov links. If we blow up a general point of degree eight on \(\mathbb P^2\) then we obtain a del Pezzo surface of degree \(1\). Such a del Pezzo surface gives a rank \(2\) fibration and an element in \(\operatorname{Cr}_2(\Bbbk)\) called a Bertini involution. This is where the hypothesis on the field \(\Bbbk\) is used. Rougly speaking the tree is constructed by recording the action of \(\operatorname{Cr}_2(\Bbbk)\) on the part of the square complex containing these Bertini involutions.
Reviewer: Shengyuan Zhao (Stony Brook)Finite groups of bimeromorphic selfmaps of uniruled Kähler threefolds.https://www.zbmath.org/1456.140202021-04-16T16:22:00+00:00"Prokhorov, Yuri G."https://www.zbmath.org/authors/?q=ai:prokhorov.yuri-g"Shramov, Constantin A."https://www.zbmath.org/authors/?q=ai:shramov.konstantin-aA group \(\Gamma\) is called Jordan if there exists a constant \(C\) such that if \(G\) is a finite subgroup of \(\Gamma\) then \(G\) has a normal abelian subgroup of index at most \(C\). The authors classified in their previous works compact complex surfaces and projective threefolds whose groups of bimeromorphic transformations are Jordan. In the paper under review they proved that the only non-algebraic compact Kähler threefolds whose groups of bimeromorphic transformations are not Jordan are projectivizations of holomorphic vector bundles of rank two over two-dimensional complex tori of algebraic dimension one. With the help of their previous results on surfaces and projective threefolds, the proof is reduced to the study of bimeromorphic self maps of conic bundles over non-algebraic surfaces.
Reviewer: Shengyuan Zhao (Stony Brook)