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Of all publications in the section: 482
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Working paper
Galkin S., Popov P. arxiv.org. math. Cornell University, 2018. No. 1810.07001.
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).
Working paper
Burman Y. M., Lvovsky S. arxiv.org. math. Cornell University, 2013. No. 1904.
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.
Working paper
Beklemishev L. D., Fernandez D., Joosten J. arxiv.org. math. Cornell University, 2012
Working paper
Prokhorov Y., Reid M. arxiv.org. math. Cornell University, 2012
Working paper
Glutsyuk A. arxiv.org. math. Cornell University, 2014. No. 1309.1843.
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.
Working paper
Viktor S. Kulikov, Shustin E. arxiv.org. math. Cornell University, 2015
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.
Working paper
Rovinsky M. arxiv.org. math. Cornell University, 2014
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).
Working paper
V'yugin V. arxiv.org. math. Cornell University, 2014. No. 1409.3865v1.
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.
Working paper
Fedor Bogomolov, Yuri Prokhorov. arxiv.org. math. Cornell University, 2013
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.
Working paper
Yuri Prokhorov. arxiv.org. math. Cornell University, 2013
We prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability.
Working paper
Lvovsky S. arxiv.org. math. Cornell University, 2013. No. 1305.2205.
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.
Working paper
Glutsyuk A., Filimonov D., Kleptsyn V. et al. arxiv.org. math. Cornell University, 2013. No. 1301.7159.
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.
Working paper
Ovcharenko M. arxiv.org. math. Cornell University, 2020
We study smooth Fano weighted complete intersections with respect to the new invariant -- the variance var(X) = coindex(X) - codim(X).
Working paper
Guseva L. arxiv.org. math. Cornell University, 2018
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.
Working paper
Vladimir L. Popov. arxiv.org. math. Cornell University, 2015. No. 1508.02860.
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.
Working paper
Rybnikov G. arxiv.org. math. Cornell University, 1998
Working paper
Vik.S. Kulikov. arxiv.org. math. Cornell University, 2014
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.
Working paper
Skripchenko A., Troubetzkoy S. arxiv.org. math. Cornell University, 2015
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.