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Working paper

Let X(n) denote n-th symmetric power of a cubic surface X. We show that X(4)×X is stably birational to X(3)×X, despite examples when X(4) is not stably birational to X(3).

Added: Oct 19, 2018

Working paper

Suppose that C⊂P2 is a general enough smooth plane curve of degree >2 and that π:C→P1 is a finite morphism simply ramified over the same set of points as a projection prp:C→P1, where p∈P2∖C. We prove that the morphism π is equivalent to such a projection if and only if it extends to a finite morphism X→(P2)∗ ramified over C∗, where X is a smooth surface. Actually we prove a similar result for nodal curves.

Added: Nov 14, 2013

Working paper

Added: Feb 12, 2013

Working paper

Added: Feb 6, 2013

Working paper

The famous conjecture of V.Ya.Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study the complex algebraic version of Ivrii's conjecture for quadrilateral orbits in two dimensions, with reflections from complex algebraic curves. We present the complete classification of 4-reflective algebraic counterexamples: billiards formed by four complex algebraic curves in the projective plane that have open set of quadrilateral orbits. As a corollary, we provide classification of the so-called real algebraic pseudo-billiards with open set of quadrilateral orbits: billiards formed by four real algebraic curves; the reflections allow to change the side with respect to the reflecting tangent line.

Added: Sep 29, 2013

Working paper

In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.

Added: Feb 2, 2015

Working paper

In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite field) are studied. Many results here are well-known to the experts, at least in the case of {\sl linear representations} of symmetric group. The presented results suggest, in particular, an analogue of Hilbert's Theorem 90 should hold: in the case of faithful action of the group on the base field the irreducible semilinear representations are one-dimensional (and trivial in appropriate sense).

Added: Sep 17, 2014

Working paper

On Stability Property of Probability Laws with Respect to Small Violations of Algorithmic Randomness

We study a stability property of probability laws with respect to
small violations of algorithmic randomness. A sufficient condition of
stability is presented in terms of Schnorr tests of algorithmic randomness.
Most probability laws, like the strong law of large numbers,
the law of iterated logarithm, and even Birkhoff’s pointwise ergodic
theorem for ergodic transformations, are stable in this sense.
Nevertheless, the phenomenon of instability occurs in ergodic theory.
Firstly, the stability property of the Birkhoff’s ergodic theorem is
non-uniform. Moreover, a computable non-ergodic measure preserving
transformation can be constructed such that ergodic theorem is
non-stable. We also show that any universal data compression scheme
is also non-stable with respect to the class of all computable ergodic
measures.

Added: Sep 17, 2014

Working paper

We discuss the problem of stable conjugacy of finite subgroups of Cremona
groups. We show that the group $H^1(G,Pic(X))$ is a stable birational invariant
and compute this group in some cases.

Added: Nov 21, 2014

Working paper

We prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability.

Added: Oct 10, 2013

Working paper

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak's classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak's theorem to the case of finite characteristic.

Added: Oct 3, 2013

Working paper

In the present paper we investigate two-parametric family of nonautonomous ordinary differential equations on the two-torus that model the Josephson effect from superconductivity. We study its rotation number as a function of parameters and its Arnold tongues (also called phase locking domains): the level sets of the rotation number that have non-empty interior. The Arnold tongues of the equation under consideration have many non-typical properties: the phase locking happens only for integer values of the rotation number; the boundaries of the tongues are given by analytic curves, the tongues have zero width at the intersection points of the latter curves (this yield the adjacency points). Numerical experiments and theoretical investigations show that each Arnold tongue forms an infinite chain of adjacent domains separated by adjacency points and going to infinity in asymptotically vertical direction. Recent numerical experiments had also shown that for each Arnold tongue all its adjacency points lie on one and the same vertical line with the integer abscissa equal to the corresponding rotation number. In the present paper we prove this fact for some open domain of the two-parametric families of equations under consideration. In the general case we prove a weaker statement: the abscissa of each adjacency point is integer; it has the same sign, as the rotation number; its modulus is no greater than that of the rotation number. The proof is based on the representation of the differential equations under consideration as projectivizations of complex linear differential equations on the Riemann sphere, see, and the classical theory of complex linear equations.

Added: May 15, 2013

Working paper

We study smooth Fano weighted complete intersections with respect to the new invariant -- the variance var(X) = coindex(X) - codim(X).

Added: Jun 12, 2020

Working paper

We construct a full exceptional collection of vector bundles in the bounded derived category of
coherent sheaves on the Grassmannian IGr(3,8) of isotropic 3-dimensional subspaces in a symplectic vector
space of dimension 8.

Added: Oct 19, 2018

Working paper

For the coordinate algebras of connected affine algebraic groups, we explore the problem of finding a presentation by generators and relations canonically determined by the group structure.

Added: Aug 13, 2015

Working paper

Added: Nov 23, 2013

Working paper

Let C_1 be an irreducible component of a reduced projective curve C⊂P^2 defined over the field C, degC_1≥2, and let T be the set of lines l⊂P^2 meeting C transversally. In the article, we prove that for a line l_0∈T and any two points P_1,P_2∈C_1∩l_0 there is a loop l_t⊂T, t∈[0,1], such that the movement of the line l_0 along the loop l_t induces the transposition of the points P_1, P_2 and the identity permutation of the other points of C∩l_0.

Added: Feb 2, 2015

Working paper

We prove linear upper and lower bounds for the Hausdorff dimension set of minimal interval exchange transformations with flips (in particular without periodic points), and a linear lower bound for the Hausdorff dimension of the set of non-uniquely ergodic minimal interval exchange transformations with flips.

Added: Nov 19, 2015

Working paper

For 3D reaction–diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic nonlinearity which does not admit a normally hyperbolic inertial manifold. An example separating the classes of such equations admitting an inertial manifold and a normally hyperbolic inertial manifold is constructed. Similar questions concerning absolutely normally hy- perbolic inertial manifolds are discussed.

Added: Jun 26, 2016

Working paper

We consider the modified Monge-Kantorovich problem with additional restriction: admissible transport plans must vanish on some fixed functional subspace. Different choice of the subspace leads to different additional properties optimal plans need to satisfy. Our main results are quite general and include several important examples. In particular, they include Monge-Kantorovich problems in the classes of invariant measures and martingales. We formulate and prove a criterion for existence of a solution, a duality statement of the Kantorovich type, and a necessary geometric condition on a support of optimal measure, which is analogues to the usual c-monotonicity.

Added: May 14, 2014

Working paper

Recently the second author and Van Proeyen proved the monodromy conjecture on topological zeta functions for all non-degenerate surface singularities. In this paper, we obtain higher-dimensional analogues of their results, which, in particular, prove the conjecture for all isolated singularities of 4 variables, as well as for many classes of non-isolated and higher-dimensional singularities. One of the tools that we introduce is a certain extension of the notion of a supermodular function, which may be of independent interest in convex analysis and game theory.

Added: Sep 18, 2017