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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.
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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.
We prove that the closure of every Jordan class J in a semisimple simply connected complex group G at a point x with Jordan decomposition x = rv is smoothly equivalent to the union of closures of those Jordan classes in the centraliser of r that are contained in J and contain x in their closure. For x unipotent we also show that the closure of J around x is smoothly equivalent to the closure of a Jordan class in Lie(G) around exp^{-1}x. For G simple we apply these results in order to determine a (non-exhaustive) list of smooth sheets in G, the complete list of regular Jordan classes whose closure is normal and Cohen-Macaulay, and to prove that all sheets and Lusztigs strata in SL(n,C) are smooth.
The first main purpose of this paper is to contribute to the existing knowledge about the complex projective surfaces $S$ of general type with $p_g(S) = 0$ and their moduli spaces, constructing 19 new families of such surfaces with hitherto unknown fundamental groups. We also provide a table containing all the known such surfaces with K^2 <=7. Our second main purpose is to describe in greater generality the fundamental groups of smooth projective varieties which occur as the minimal resolutions of the quotient of a product of curves by the action of a finite group. We classify, in the two dimensional case, all the surfaces with q=p_g = 0 obtained as the minimal resolution of such a quotient, having rational double points as singularities. We show that all these surfaces give evidence to the Bloch conjecture.
We prove that an analogue of Jordans theorem on finite subgroups of general linear groups does not hold for the group of bimeromorphic automorphisms of a product of the complex projective line and a complex torus of positive algebraic dimension.
We prove a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of $G$ and the group $G(k)$ of $k$-points. For $G$ simple and $V$ faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those $G$ and $V$ for which the stabilizer in general position is smooth, or $dim V/G < dim G$, or there is a $v in V$ whose stabilizer in $G$ is trivial.
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