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## Generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes

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In this paper we prove that the generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes of points in the plane has a beautiful decomposition into an infinite product. We also compute the generating series of the numbers of quasihomogeneous components in a moduli space of sheaves on the projective plane. The answer is given in terms of characters of the affine Lie algebra~$\widehat{sl}_m$.

### In book

Vol. 40: Symmetries, Integrable Systems and Representations. , Springer, 2013

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

Galkin S., Shinder E., The Fano variety of lines and rationality problem for a cubic hypersurface / 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., On pairs, triples and quadruples of points on a cubic surface / 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

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

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

Kuznetsov A., Prokhorov Y., Shramov K., Hilbert schemes of lines and conics and automorphism groups of Fano threefolds / 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

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

Buryak A., Moscow Mathematical Journal 2012 Vol. 12 No. 1 P. 1-17

In this paper we give a formula for the classes (in the Grothendieck ring of complex quasi-projective varieties) of irreducible components of (1,k)-quasi-homogeneous Hilbert schemes of points on the plane. We find a new simple geometric interpretation of the q,t-Catalan numbers. Finally, we investigate a connection between (1,k)-quasi-homogeneous Hilbert schemes and homogeneous nested Hilbert schemes. ...

Added: October 1, 2020

Popov P., Twisted cubics and quadruples of points on cubic surfaces / 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

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

Pavlov A., Proceedings of the American Mathematical Society 2020 Vol. 148 No. 4 P. 1373-1381

Let X be a smooth projective Calabi-Yau variety and let L be a Koszul line bundle on X. We show that for Betti numbers of a maximal Cohen-Macaulay module over the homogeneous coordinate ring A of X there are formulas similar to the formulas for cohomology numbers. This similarity is realized via the box-product resolution of ...

Added: October 31, 2020

Gorsky E., Negut A., Journal de Mathématiques Pures and Appliquées 2015 Vol. 104 No. 3 P. 403-435

We consider the construction of refined Chern-Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We prove Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants with the Hilbert scheme of points on the ...

Added: February 14, 2015

Kurnosov N., Mathematical notes 2016 Vol. 99 No. 1 P. 330-334

Added: June 8, 2016

Gorsky E., Hogancamp M., Hilbert schemes and y-ification of Khovanov-Rozansky homology / 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

Gorsky E., 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

Pavlov A., Journal of Algebra 2019 Vol. 526 P. 211-242

We apply Orlov's equivalence to derive formulas for the Betti numbers of maximal Cohen-Macaulay modules over the cone an elliptic curve $(E,x)$ embedded into $\mathbb{P}^{n-1}$, by the full linear system $|\mathcal{O}(nx)|$, for $n>3$. The answers are given in terms of recursive sequences. These results are applied to give a criterion of (Co-)Koszulity.
In the last two ...

Added: May 24, 2019

Gorsky E., 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

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

Pavlov A., Mathematische Zeitschrift 2021 No. 297 P. 223-254

We show that for maximal Cohen–Macaulay modules over the homogeneous coordinate ring of a smooth Calabi–Yau varieties X, the computation of Betti numbers can be reduced to computations of dimensions of certain HomHom spaces in the bounded derived category Db(X). In the simplest case of a smooth elliptic curve E embedded in P2 as a smooth cubic, we get explicit values for Betti ...

Added: October 31, 2020

Buryak A., Feigin B. L., Nakajima H., International Mathematics Research Notices 2015 No. 13

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: October 10, 2015

Gorsky E., Negut A., Rasmussen J., Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology / Cornell University. Series arXiv "math". 2016.

We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of ...

Added: September 19, 2016