On stable conjugacy of finite subgroups of the plane Cremona group, I
We discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We compute the stable birational invariant H 1(G, Pic(X)) for cyclic groups of prime order.
Co-fibered products of algebraic curves
We give examples of failure of the existence of co-fibered products in the category of algebraic curves.
Unramified cohomology of alternating groups
We prove vanishing results for the unramified stable cohomology of alternating groups.
On a theorem of Tate
We study applications of divisibility properties of recurrence sequences to Tate’s theory of abelian varieties over finite fields.
On the irreducibility of Hilbert scheme of surfaces of minimal degree
The article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙm+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof of Chisini conjecture [Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian)] for coverings of projective plane branched in a special class of rational curves.
Stable cohomology of alternating groups
We determine the stable cohomology groups ( of the alternating groups for all integers n and i, and all odd primes p.
Collineation group as a subgroup of the symmetric group
Let ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let H be a closed (in the pointwise convergence topology) subgroup of the permutation group of the set ψ. Suppose that H contains the projective group and an arbitrary self-bijection of ψ transforming a triple of collinear points to a non-collinear triple. It is well known from [Kantor W.M., McDonough T.P., On the maximality of PSL(d+1,q), d ≥ 2, J. London Math. Soc.,...
On the diffeomorphic type of the complement to a line arrangement in a projective plane
We show that the diffeomorphic type of the complement to a line arrangement in a complex projective plane P 2 depends only on the graph of line intersections if no line in the arrangement contains more than two points in which at least two lines intersect. This result also holds for some special arrangements which do not satisfy this property. However it is not true in general, see [Rybnikov G., On the fundamental group of the complement of a complex hyperplane arrangement, Funct. Anal. Appl., 2011,...
Ordinary reduction of K3 surfaces
Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.
A version of Brauer’s theorem for integer central functions
In this article we prove an effective version of the classical Brauer’s Theorem for integer class functions on finite groups.
Singularities on complete algebraic varieties
We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety.
Linear bounds for levels of stable rationality
Let G be one of the groups SLn(ℂ), Sp2n (ℂ), SOm(ℂ), Om(ℂ), or G 2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G × ℙN is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G being one of the classical groups.
Obituary: Vasyl Ivanovych Andriychuk (18.09.1948–7.07.2012)
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