Related papers: Cubic surface bundles and the Brauer group

This is an English translation of the author's 1989 note in Russian, published in a collection "Arithmetic and Geometry of Varieties" (V.E. Voskresenski, ed.), Kuibyshev State University, Kuibyshev, 1989, pp. 57--67. Let $X$ be be an…

Let $G$ be a semisimple linear algebraic group over the field $\mathbb C$, and let $C$ be an irreducible smooth complex projective curve of genus at least three. We compute the Brauer group of the smooth locus of the moduli space of…

We study subgroups of ${\rm PU}(2,1)$ generated by two non-commuting unipotent maps $A$ and $B$ whose product $AB$ is also unipotent. We call $\mathcal{U}$ the set of conjugacy classes of such groups. We provide a set of coordinates on…

Let $S$ be a regular surface endowed with a very ample line bundle $\mathcal O_S(h_S)$. Taking inspiration from a very recent result by D. Faenzi on $K3$ surfaces, we prove that if $\mathcal O_S(h_S)$ satisfies a short list of technical…

We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible…

We construct a natural branch divisor for equidimensional projective morphisms where the domain has lci singularities and the target is nonsingular. The method involves generalizing a divisor contruction of Mumford from sheaves to…

We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group…

We prove that the complex surfaces parametrizing cuboids and face cuboids, as well as their minimal resolution of singularities, have trivial fundamental group. We then compute the fundamental group of certain open smooth subvarieties of…

The goal of this work is to give new quantitative results about the distribution of semi-arithmetic hyperbolic surfaces in the moduli space of closed hyperbolic surfaces. We show that two coverings of genus $g$ of a fixed arithmetic surface…

We provide a differential structure on arbitrary cleft extensions $B:=A^{\mathrm{co}H}\subseteq A$ for an $H$-comodule algebra $A$. This is achieved by constructing a covariant calculus on the corresponding crossed product algebra…

We compute explicit bases for the de Rham cohomology of cyclic covers of the projective line defined over an algebraically closed field of characteristic $p\geq 0$. For both Kummer and Artin-Schreier extensions, we describe precise…

We introduce a category of noncommutative bundles. To establish geometry in this category we construct suitable noncommutative differential calculi on these bundles and study their basic properties. Furthermore we define the notion of a…

We use constructions of surfaces as abelian covers to write down exceptional collections of line bundles of maximal length for every surface $X$ in certain families of surfaces of general type with $p_g=0$ and $K_X^2=3,4,5,6,8$. We also…

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with…

We introduce the notion of $\pi^2$-graded Hopf algebra, where the grading is by the double groupoid of commutative diagrams of a finite groupoid $\pi$. The finite dimensional representations of a $\pi^2$-graded Hopf algebra form a rigid…

We classify finite groups acting by birational transformations of a non-trivial Severi--Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism…

In this paper, we give a simple and geometric, but formal, description of an open subset of the character variety of surface groups into $\mathrm{SL}_n(\mathbb{C})$. The main ingredient is a modified version of the WKB method, which we call…

We study Brauer-Severi surface bundles over smooth projective varieties via root stacks, with a view towards applications to failure of stable rationality.

We show that for any ample line bundle on a smooth complex projective variety with nonnegative Kodaira dimension, the semistability of co-Higgs bundles of implies the semistability of bundles. Then we investigate the criterion for surface…

Deformation quantization algebroids over a complex symplectic manifold X are locally given by rings of WKB operators, that is, microdifferential operators with an extra central parameter \tau. In this paper, we will show that such…