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## On pairs, triples and quadruples of points on a cubic surface

Cornell University
,
2018.
No. 1810.07001.

Galkin S., Popov P.

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).

Galkin S., Shinder E., / Cornell University. Series math "arxiv.org". 2014. No. 1405.5154.

We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational ...

Added: May 21, 2014

Popov P., / Cornell University. Series math "arxiv.org". 2018. No. 1810.04563.

We study relations in the Grothendieck ring of varieties which connect the Hilbert scheme of points on a cubic hypersurface Y with a certain moduli space of twisted cubic curves on Y. These relations are generalizations of the "beautiful" Y-F(Y) relation by Galkin and Shinder which connects Y with the Hilbert scheme of two points on Y and the Fano variety F(Y) of lines on Y. We ...

Added: October 23, 2018

Prokhorov Y., Annales de l'Institut Fourier 2015 No. 65 P. 1–16

We prove that for a Q-Gorenstein degeneration $X$ of del Pezzo surfaces, the number of non-Du Val singularities is at most $\rho(X)+2$. Degenerations with $\rho(X)+2$ and $\rho(X)+1$ non-Du Val points are investigated. ...

Added: October 17, 2014

Kishimoto T., Yuri Prokhorov, Zaidenberg M., Algebraic Geometry 2014 Vol. 1 No. 1 P. 46–56

In a previous paper we established that for any del Pezzo surface Y of degree at least 4, the affine cone X over Y embedded via a pluri-anticanonical linear system admits an effective Ga-action. In particular, the group Aut(X) is infinite dimensional. In contrast, we show in this note that for a del Pezzo surface ...

Added: October 10, 2013

Yuri Prokhorov, Documenta Mathematica 2010 Vol. 15 P. 843–872

We study Q-Fano threefolds of large Fano index. In
particular, we prove that the maximum possible Fano index is attained
only by the weighted projective space P(3,4,5,7). ...

Added: December 6, 2013

Rovinsky M., / Cornell University. Series math "arxiv.org". 2012.

I show that the cohomology of the generic points of algebraic complex varieties becomes stable birational invariant, when considered `modulo the cohomology of the generic points of the affine spaces'. ...

Added: October 31, 2013

Prokhorov Y., Kuznetsov A., Shramov K., Japanese Journal of Mathematics 2018 Vol. 13 No. 1 P. 109–185

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds of Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold ...

Added: November 22, 2017

Kishimoto T., Yuri Prokhorov, Zaidenberg M., Transformation Groups 2013 Vol. 18 No. 4 P. 1137–1153

We give a criterion of existence of a unipotent group action on the affine cone over a projective variety or, more generally, on the affine quasicone over a variety which is projective over another affine variety. ...

Added: October 10, 2013

Yuri Prokhorov, / Cornell University. Series math "arxiv.org". 2011.

We prove that for a Q-Gorenstein degeneration $X$ of del Pezzo surfaces, the number of non-Du Val singularities is at most $\rho(X)+2$. Degenerations with $\rho(X)+2$ and $\rho(X)+1$ non-Du Val points are investigated. ...

Added: October 11, 2013

Vologodsky V., Finkelberg M. V., Bezrukavnikov R., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163–190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 17, 2015

Kishimoto T., Yuri Prokhorov, Zaidenberg M., Osaka Journal of Mathematics 2014 Vol. 51 No. 4 P. 1093–1113

We address the following question: When an affine cone over a smooth Fano threefold admits an effective action of the additive group? In this paper we deal with Fano threefolds of index 1 and Picard number 1. Our approach is based on a geometric criterion from our previous paper, which relates the existence of an ...

Added: October 10, 2013

Bezrukavnikov R., Finkelberg M. V., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163–190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 20, 2014

Prokhorov Y., Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 215–229

We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds. ...

Added: January 24, 2014

Glutsyuk A., / Cornell University. Series math "arxiv.org". 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 ...

Added: September 29, 2013

Eugene Gorsky, Oblomkov A., Rasmussen J. et al., Duke Mathematical Journal 2014 Vol. 163 No. 14 P. 2709–2794

We conjecturally extract the triply graded Khovanov–Rozansky homology of the (m,n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n-1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the ...

Added: December 9, 2014

Buryak A., Feigin B. L., Nakajima H., International Mathematics Research Notices 2015 Vol. 2015 No. 13 P. 4708–4715

In a recent paper the first two authors proved that the generating series of the Poincare polynomials of the quasihomogeneous Hilbert schemes of points in the plane has a simple decomposition in an infinite product. In this paper we give a very short geometrical proof of that formula. ...

Added: September 29, 2020

Prokhorov Y., Shramov K., Mathematical Research Letters 2018 Vol. 25 No. 3 P. 957–972

We classify threefolds with non-Jordan birational automorphism groups. ...

Added: October 4, 2018

Rybakov S., Trepalin A., Математический сборник 2017 Т. 208 № 9 С. 148–170

Пусть X -- минимальная поверхность над полем F_q. Образ Г группы Галуа Gal ( \bar{F_q}, F_q ) в группе автоморфизмов Aut ( Pic X ) является циклической подгруппой группы Вейля W ( E_6 ). В этой подгруппе 25 классов сопряженности циклических подгрупп, и пять из них соответствуют минимальным кубическим поверхностям. Возникает естественный вопрос: какие классы сопряженности ...

Added: October 23, 2017

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2013. No. 1307.5522.

This is an expanded version of my talk at the workshop ``Groups of Automorphisms in Birational and Affine Geometry'', October 29–November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. ...

Added: July 21, 2013

Bogomolov F. A., Kulikov V. S., Central European Journal of Mathematics 2013 Vol. 11 No. 2 P. 254–263

The article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙ m+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 ...

Added: November 21, 2012

Prokhorov Y., Mori S., Известия РАН. Серия математическая 2019 Т. 83 № 3 С. 158–212

Росток экстремальной окрестности – это аналитический росток трехмерного многообразия с терминальными особенностями вдоль приведенной полной кривой, допускающий стягивание, слои которого не более чем одномерны. Цель настоящей статьи – дать обзор результатов, касающихся стягиваний с неприводимым центральным слоем, содержащих только одну негоренштейнову точку. ...

Added: June 4, 2019

Rovinsky M., Moscow Mathematical Journal 2015 Vol. 15 No. 4 P. 777–803

I show that the cohomology of the generic points of algebraic complex varieties becomes stable birational invariant, when considered ‘modulo the cohomology of the generic points of the affine spaces’. ...

Added: October 14, 2015

Gorsky Evgeny, Mazin M., Journal of Combinatorial Theory, Series A 2013 Vol. 120 No. 1 P. 49–63

J. Piontkowski described the homology of the Jacobi factor of a plane curve singularity with one Puiseux pair. We discuss the combinatorial structure of his answer, in particular, relate it to the bigraded deformation of Catalan numbers introduced by A. Garsia and M. Haiman. ...

Added: December 9, 2014

Gorsky E., Geometry and Topology 2018 Vol. 22 P. 645–691

We conjecture an expression for the dimensions of the Khovanov–Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture (2012) relating the HOMFLY polynomial to the Euler numbers of the ...

Added: August 21, 2018