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A group $G$ is called Jordan if there is a positive integer $J=J_G$ such that every finite subgroup $mathcal{B}$ of $G$ contains a commutative subgroup $mathcal{A}subset mathcal{B}$ such that $mathcal{A}$ is normal in $mathcal{B}$ and the index $[mathcal{B}:mathcal{A}] le J$ (V.L. Popov). In this paper we deal with Jordaness properties of the groups $Bir(X)$ of birational automorphisms of irreducible smooth projective varieties $X$ over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov - C. Shramov) that $Bir(X)$ is Jordan if $X$ is non-uniruled. On the other hand, the second named author proved that $Bir(X)$ is not Jordan if $X$ is birational to a product of the projective line and a positive-dimensional abelian variety. We prove that $Bir(X)$ is Jordan if (uniruled) $X$ is a conic bundle over a non-uniruled variety $Y$ but is not birational to a product of $Y$ and the projective line. (Such a conic bundle exists only if $dim(Y)ge 2$.) When $Y$ is an abelian surface, this Jordaness property result gives an answer to a question of Prokhorov and Shramov.
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We prove that an analogue of Jordans theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of algebraic surfaces. This gives a positive answer to a question of Vladimir L. Popov.
We study the groups of biholomorphic and bimeromorphic automorphisms of conic bundles over certain compact complex manifolds of algebraic dimension zero.
We deal with some pcf investigations mostly motivated by abelian group theory problems and deal their applications to test problems (we expect reasonably wide applications). We prove almost always the existence of aleph_omega-free abelian groups with trivial dual, i.e. no non-trivial homomorphisms to the integers. This relies on investigation of pcf; more specifically, for this we prove that almost always there are F subseteq lambda^kappa which are quite free and has black boxes. The almost always means that there are strong restrictions on cardinal arithmetic if the universe fails this, the restrictions are everywhere. Also we replace Abelian groups by R-modules, so in some sense our advantage over earlier results becomes clearer.
A conic bundle is a contraction $Xto Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov which predicts that, if $Xto Z$ is a conic bundle such that $X$ has canonical singularities and $Z$ is $mathbb{Q}$-Gorenstein, then $Z$ is always $frac{1}{2}$-lc, and the multiplicities of the fibers over codimension $1$ points are bounded from above by $2$. Both values $frac{1}{2}$ and $2$ are sharp. This is achieved by solving a more general conjecture of Shokurov on singularities of bases of lc-trivial fibrations of relative dimension $1$ with canonical singularities.
It goes back to Ahlfors that a real algebraic curve admits a real-fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, we are interested in characterising real algebraic varieties of dimension $n$ admitting real-fibered morphisms to the $n$-dimensional projective space. We present a criterion to classify real-fibered morphisms that arise as finite surjective linear projections from an embedded variety which relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real-fibered morphisms from real del Pezzo surfaces to the projective plane and determine which such morphisms arise as the composition of a projective embedding with a linear projection. Furthermore, we give some insights in the case of real conic bundles.
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