scholarly journals Generating Series of the Poincaré Polynomials of Quasihomogeneous Hilbert Schemes

2013 ◽  
pp. 15-33 ◽  
Author(s):  
A. Buryak ◽  
B. L. Feigin
Keyword(s):  
Hilbert Schemes ◽  
Generating Series

On Generating Series of Classes of Equivariant Hilbert Schemes of Fat Points

2010 ◽  
Vol 10 (3) ◽  
pp. 593-602 ◽  
Author(s):  
S. Gusein-Zade ◽  
I. Luengo ◽  
A. Melle-Hernández
Keyword(s):  
Hilbert Schemes ◽  
Fat Points ◽  
Generating Series

scholarly journals Generating Series in the Cohomology of Hilbert Schemes of Points on Surfaces

2007 ◽  
Vol 10 ◽  
pp. 254-270 ◽  
Author(s):  
Samuel Boissière ◽  
Marc A. Nieper-Wisskirchen
Keyword(s):  
Smooth Surface ◽  
Vertex Algebra ◽  
Characteristic Classes ◽  
Vertex Operators ◽  
Chern Classes ◽  
Hilbert Schemes ◽  
Rewriting Logic ◽  
Rational Cohomology ◽  
Generating Series ◽  
Logic System

In the study of the rational cohomology of Hilbert schemes of points on a smooth surface, it is particularly interesting to understand the characteristic classes of the tautological bundles and the tangent bundle. In this note we pursue this study. We first collect all results appearing separately in the literature and prove some new formulas using Ohmoto's results on orbifold Chern classes on Hilbert schemes. We also explain the algorithmic counterpart of the topic: the cohomology space is governed by a vertex algebra that can be used to compute characteristic classes. We present an implementation of the vertex operators in the rewriting logic system MAUDE, and address observations and conjectures obtained after symbolic computations.

scholarly journals Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points

2006 ◽  
Vol 54 (2) ◽  
pp. 353-359 ◽  
Author(s):  
S. M. Gusein-Zade ◽  
I. Luengo ◽  
A. Melle-Hernández
Keyword(s):  
Power Structure ◽  
Hilbert Schemes ◽  
Grothendieck Ring ◽  
Generating Series

scholarly journals Generating series of classes of Hilbert schemes of points on orbifolds

2009 ◽  
Vol 267 (1) ◽  
pp. 125-130
Author(s):  
S. M. Gusein-Zade ◽  
I. Luengo ◽  
A. Melle-Hernández
Keyword(s):  
Hilbert Schemes ◽  
Generating Series

scholarly journals Topology of $\mathbb{Z}_3$-Equivariant Hilbert Schemes

10.37236/8290 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Deborah Castro ◽  
Dustin Ross
Keyword(s):  
Betti Numbers ◽  
Product Formula ◽  
Hilbert Schemes ◽  
Generating Series

Motivated by work of Gusein-Zade, Luengo, and Melle-Hernández, we study a specific generating series of arm and leg statistics on partitions, which is known to compute the Poincaré polynomials of $\mathbb{Z}_3$-equivariant Hilbert schemes of points in the plane, where $\mathbb{Z}_3$ acts diagonally. This generating series has a conjectural product formula, a proof of which has remained elusive over the last ten years. We introduce a new combinatorial correspondence between partitions of $n$ and $\{1,2\}$-compositions of $n$, which behaves well with respect to the statistic in question. As an application, we use this correspondence to compute the highest Betti numbers of the $\mathbb{Z}_3$-equivariant Hilbert schemes.

scholarly journals Curved Rickard complexes and link homologies

2020 ◽  
Vol 2020 (769) ◽  
pp. 87-119
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Joshua Sussan
Keyword(s):  
Spectral Sequence ◽  
Quantum Groups ◽  
Braid Group ◽  
Derived Categories ◽  
Duke Math ◽  
Khovanov Homology ◽  
Hilbert Schemes ◽  
Coherent Sheaves ◽  
Knot Homology ◽  
Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

scholarly journals A symmetric Bloch–Okounkov theorem

2021 ◽  
Vol 8 (2) ◽  
Author(s):  
Jan-Willem M. van Ittersum
Keyword(s):  
Symmetric Functions ◽  
Graded Algebra ◽  
Symmetric Polynomials ◽  
Generating Series

AbstractThe algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.

scholarly journals Holomorphic anomaly equation for and the Nekrasov-Shatashvili limit of local

2021 ◽  
Vol 9 ◽  
Author(s):  
Pierrick Bousseau ◽  
Honglu Fan ◽  
Shuai Guo ◽  
Longting Wu
Keyword(s):  
Elliptic Curve ◽  
Moduli Spaces ◽  
Topological String ◽  
Holomorphic Anomaly ◽  
Holomorphic Anomaly Equation ◽  
Anomaly Equation ◽  
Witten Theory ◽  
Generating Series ◽  
Maximal Contact ◽  
Higher Genus

Abstract We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $\lambda _g$ -insertion is related to Gromov-Witten theory of the total space of ${\mathcal O}_X(-D)$ and local Gromov-Witten theory of D. Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${\mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to $S={\mathbb P}^2$ , we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${\mathbb P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${\mathbb P}^2$ , we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${\mathbb P}^2$ in the Nekrasov-Shatashvili limit.

scholarly journals The close relation between border and Pommaret marked bases

2021 ◽  
Author(s):  
Cristina Bertone ◽  
Francesca Cioffi
Keyword(s):  
Finite Order ◽  
Polynomial Ring ◽  
Group Actions ◽  
Close Relation ◽  
Open Covering ◽  
Order Ideal ◽  
Lower Dimension ◽  
Hilbert Schemes ◽  
Affine Spaces ◽  
The Ideal

AbstractGiven a finite order ideal $${\mathcal {O}}$$ O in the polynomial ring $$K[x_1,\ldots , x_n]$$ K [ x 1 , … , x n ] over a field K, let $$\partial {\mathcal {O}}$$ ∂ O be the border of $${\mathcal {O}}$$ O and $${\mathcal {P}}_{\mathcal {O}}$$ P O the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$ O . In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$ ∂ O -marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$ ∂ O -marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $$\partial {\mathcal {O}}$$ ∂ O -marked bases and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.

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