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Twisted cubics and quadruples of points on cubic surfaces
Cornell University
,
2018.
No. 1810.04563.
Popov P.
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 concentrate mostly on the case of cubic surfaces. The symmetries of 27 lines on a smooth cubic surface give a lot of restrictions on possible forms of the relations.
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
Galkin S., Popov P., / Cornell University. Series math "arxiv.org". 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). ...
Added: October 19, 2018
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
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., 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
Gusein-Zade S., Proceedings of the Edinburgh Mathematical Society 2019 Vol. 62 No. 4 P. 925-948
We define a Grothendieck ring of varieties with actions of finite groups and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural λ-structures on the ring and the corresponding power structures over it and ...
Added: October 27, 2020
Gusein-Zade S., Revista Matemática Complutense 2018 Vol. 31 No. 3 P. 595-609
A power structure over a ring is a method to give sense to expressions of the form $(1+a_1t+a_2t^2...)^m$, where $a_i$, $i=1,2, ...$, and $m$ are elements of the ring. The (natural) power structure over the Grothendieck ring of complex quasi-projective varieties appeared to be useful for a number of applications. We discuss new examples of ...
Added: October 27, 2020
Galkin S., Belmans P., Mukhopadhyay S., / Cornell University. Series math "arxiv.org". 2020. No. 2009.05568.
We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. These graphs encode degenerations of curves to rational curves, and graph potentials encode degenerations of the moduli space of rank 2 bundles with fixed determinant. We show that the birational type of the graph potential only depends on the homotopy type of ...
Added: April 15, 2021
Perepechko A., Forum Mathematicum 2021 Vol. 33 No. 2 P. 339-348
Let Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on ...
Added: January 15, 2021
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
Kuznetsov A., Prokhorov Y., Shramov K., / Cornell University. Series arXiv "math". 2016.
We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with 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: May 16, 2016
Gorsky E., Hogancamp M., / Cornell University. Series arXiv "math". 2017.
We define a deformation of the triply graded Khovanov-Rozansky homology of a link L depending on a choice of parameters for each component of L. We conjecture that this invariant restores the missing symmetry of the triply graded Khovanov-Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection with Hilbert schemes of points ...
Added: December 28, 2017
Buryak A., Feigin B. L., Nakajima H., International Mathematical 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
Kuznetsov A., Shinder E., / Cornell University. Series arXiv "math". 2016.
We discuss a conjecture saying that derived equivalence of smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection ...
Added: April 10, 2017
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
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
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
Trepalin A., European Journal of Mathematics 2016 Vol. 2 No. 1 P. 333-359
Let 𝕜 be any field of characteristic zero, X be a cubic surface in ℙ3 and G be a group acting on X. We show that if X(𝕜)≠∅ and G is not trivial and not a group of order 3 acting in a special way then the quotient surface X/G is rational over 𝕜. For ...
Added: October 22, 2015
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
Kuznetsov A., Shinder E., Selecta Mathematica, New Series 2018 Vol. 24 No. 4 P. 3475-3500
We discuss a conjecture saying that derived equivalence of simply connected smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth ...
Added: June 14, 2017
Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189
The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...
Added: January 28, 2020
Borzykh D., ЛЕНАНД, 2021
Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...
Added: February 20, 2021
В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80
Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...
Added: May 3, 2017
Красноярск : ИВМ СО РАН, 2013
Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...
Added: November 18, 2013