We introduce and study a semigroup structure on the set of irreducible components of the Hurwitz space of marked coverings of a complex projective curve with given Galois group of the coverings and fixed ramification type. As application, we give new conditions on the ramification type that are sufficient for irreducibility of the Hurwitz spaces, suggest some bounds on the number of irreducibility components under certain more general conditions, and show that the number of irreducible components coincides with the number of topological classes of the coverings if the number of brunch points is big enough.

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Working paper

We compute the class which is Poincare dual to the first Stiefel-Whitney class for the Deligne-Mumford compactification of the moduli space of real algebraic curves of genus 0 with n marked and numbered points in terms of the natural cell decomposition of the variety under consideration.

Added: Oct 19, 2014

Working paper

The paper addresses the tolerance approach to the sensitivity analysis of optimal solutions to a nonlinear discrete
optimization problem, which involves a continuous, associative, commutative,
nondecreasing and unbounded binary operation of generalized addition on nonnegative reals, called an A-operation. We
evaluate and present sharp estimates for upper and lower bounds of costs of elements from the ground set, for which an
optimal solution to the above problem remains stable. These bounds present new results in the sensitivity
analysis as well as extend most known results in a unified way. We define an invariant of the optimization
problem—the tolerance function, which is independent of optimal solutions, and establish its basic properties,
among which we mention a characterization of the set of all optimal solutions, the uniqueness of optimal
solutions and extremal values of the tolerance function on an optimal solution.

Added: Dec 13, 2013

Working paper

We present a modification of the method of conical resolutions \cite{quintics,tom}. We apply our construction to compute the rational cohomology of the spaces of equations of nodal cubics in CP2, nodal quartics in CP2 and nodal cubics in CP3. In the last two cases we also compute the cohomology of the corresponding moduli spaces.

Added: Feb 26, 2014

Working paper

We study some extension of a discrete Heisenberg group coming from the theory of loop-groups and find invariants of conjugacy classes in this group. In some cases including the case of the integer Heisenberg group we make these invariants more explicit.

Added: Nov 14, 2014

Working paper

Contraherent cosheaves are globalizations of cotorsion (or similar) modules over commutative rings obtained by gluing together over a scheme. The category of contraherent cosheaves over a scheme is a Quillen exact category with exact functors of infinite product. Over a quasi-compact semi-separated scheme or a Noetherian scheme of finite Krull dimension (in a different version - over any locally Noetherian scheme), it also has enough projectives. We construct the derived co-contra correspondence, meaning an equivalence between appropriate derived categories of quasi-coherent sheaves and contraherent cosheaves, over a quasi-compact semi-separated scheme and, in a different form, over a Noetherian scheme with a dualizing complex. The former point of view allows us to obtain an explicit construction of Neeman's extraordinary inverse image functor f^! for a morphism of quasi-compact semi-separated schemes f. The latter approach provides an expanded version of the covariant Serre-Grothendieck duality theory and leads to Deligne's extraordinary inverse image functor f^! (which we denote by f^+) for a morphism of finite type f between Noetherian schemes. Semi-separated Noetherian stacks, affine Noetherian formal schemes, and ind-affine ind-schemes (together with the noncommutative analogues) are briefly discussed in the appendices.

Added: Feb 6, 2013

Working paper

Pointwise convergence of spherical averages is proved for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen-Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving a reversible Markov chain to which methods previously introduced by the first author for the case of free groups may be applied.

Added: Sep 18, 2018

Working paper

In this paper, we consider projection estimates for L{\'e}vy densities in high-frequency setup. We give a unified treatment for different sets of basis functions and focus on the asymptotic properties of the maximal deviation distribution for these estimates. Our results are based on the idea to reformulate the problems in terms of Gaussian processes of some special type and to further analyze these Gaussian processes. In particular, we construct a sequence of excursion sets, which guarantees the convergence of the maximal deviation distribution to the Gumbel distribution. We show that the rates of convergence presented in previous research on this topic are logarithmic and construct the sequences of accompanying laws with power rates.

Added: Dec 12, 2014

Working paper

Following an old suggestion of M. Kontsevich, and inspired by recent work of A. Beilinson and B. Bhatt, we introduce a new version of periodic cyclic homology for DG agebras and DG categories. We call it co-periodic cyclic homology. It is always torsion, so that it vanishes in char 0. However, we show that co-periodic cyclic homology is derived-Morita invariant, and that it coincides with the usual periodic cyclic homology for smooth cohomologically bounded DG algebras over a torsion ring. For DG categories over a field of odd positive characteristic, we also establish a non-commutative generalization of the conjugate spectral sequence converging to our co-periodic cyclic homology groups.

Added: Nov 20, 2015

Working paper

Added: Sep 10, 2012

Working paper

Added: Feb 6, 2013

Working paper

On del Pezzo surfaces, we study effective ample R-divisors such that the complements of their supports are isomorphic to A1-bundles over smooth affine curves.

Added: Nov 18, 2015

Working paper

For each del Pezzo surface $S$ with du Val singularities, we determine
whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we
present an effective divisor $D$ that is $\mathbb{Q}$-linearly equivalent to
$-K_S$ and such that the open set $S\setminus\mathrm{Supp}(D)$ is a cylinder.
As a corollary, we classify all the del Pezzo surfaces with du Val
singularities that admit nontrivial $\mathbb{G}_a$-actions on their affine
cones.

Added: Dec 27, 2013

Working paper

We consider L^p-Wasserstein distances on a subset of probability measures. If the subset of interest appears to be a simplex, these distances are determined by their values on extreme points of the simplex. We show that this fact is a corollary of the following decomposition result: an optimal transport plan can be represented as a mixture of optimal transport plans between extreme points of the simplex. This fact can be generalized to the Kantorovich problem with additional linear restriction and the associated Wasserstein-like distances. We prove that the decomposition is possible, if marginal measures are elements of a simplex that is compatible with the additional restriction.

Added: May 25, 2015

Working paper

The description of global deformations of Lie algebras is important since it is related to unsolved problem of the classification of simple Lie algebras over a field of small characteristic.
In this paper we study global deformations of Lie algebras of type ${D}_{l}$ over an algebraically closed field K of characteristic 2. It is proved that Lie algebras of type $\bar{D_{l}}$
are rigid for odd $l>3$. Some global deformations of Lie algebras of type ${D}_{l}$ are constructed for even $l\ge 4$.

Added: Dec 8, 2017

Working paper

We study the connection between the affine degenerate Grassmannians in type $A$, quiver Grassmannians for one vertex loop quivers and affine
Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type $GL_n$ and identify it
with semi-infinite orbit closure of type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex loop quiver
provide finite-dimensional approximations of the degenerate affine Grassmannian. Finally, we give an explicit
description of the degenerate affine Grassmannian of type $A_1^{(1)}$, propose
a conjectural description in the symplectic case and discuss the generalization to the case of the affine degenerate flag varieties.

Added: Oct 6, 2014

Working paper

Added: Feb 6, 2013

Working paper

Let M be a hyperkaehler manifold, and η a closed, positive (1,1)-form which is degenerate everywhere on M. We associate to η a family of complex structures on M, called a degenerate twistor family, and parametrized by a complex line. When η is a pullback of a Kaehler form under a Lagrangian fibration L, all the fibers of degenerate twistor family also admit a Lagrangian fibration, with the fibers isomorphic to that of L. Degenerate twistor families can be obtained by taking limits of twistor families, as one of the Kahler forms in the hyperkahler triple goes to η.

Added: Dec 27, 2013

Working paper

We obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain one-parameter degenerations of the Lie algebras corresponding to such subgroups. As an application, we exhibit an explicit algorithm for computing the set of spherical roots of such a spherical subgroup.

Added: Jun 1, 2019

Working paper

Given a singular variety I discuss the relations between quantum cohomology of its resolution and smoothing. In particular, I explain how toric degenerations helps with computing Gromov--Witten invariants, and the role of this story in Fanosearch programme. The challenge is to formulate enumerative symplectic geometry of complex 3-folds in a way suitable for extracting invariants under blowups, contractions, and transitions.

Added: Sep 25, 2018

Working paper

I prove new local inequality for divisors on smooth surfaces, describe its applications, and compare it to a similar local inequality that is already known by experts.

Added: Dec 27, 2013

Working paper

We estimate δ-invariants of some singular del Pezzo surfaces with quotient singularities, which we studied ten years ago. As a result, we show that each of these surfaces admits an orbifold K\"ahler--Einstein metric.

Added: Oct 21, 2018