Generating series of intersection numbers on Hilbert schemes of points

2017 ◽  
Vol 12 (5) ◽  
pp. 1247-1264 ◽  
Author(s):  
Zhilan Wang ◽  
Jian Zhou
Keyword(s):  
Hilbert Schemes ◽  
Intersection Numbers ◽  
Generating Series

scholarly journals Intersection numbers on the relative Hilbert schemes of points on surfaces

2017 ◽  
Vol 21 (3) ◽  
pp. 531-542 ◽  
Author(s):  
Amin Gholampour ◽  
Artan Sheshmani
Keyword(s):  
Hilbert Schemes ◽  
Intersection Numbers

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 Stable Pair Invariants of Local Calabi–Yau 4-folds

2021 ◽  
Author(s):  
Yalong Cao ◽  
Martijn Kool ◽  
Sergej Monavari
Keyword(s):  
Line Bundles ◽  
Zero Section ◽  
Hilbert Schemes ◽  
Intersection Numbers ◽  
Genus Zero ◽  
Stable Pair ◽  
Local Principle ◽  
Low Degree ◽  
Witten Theory ◽  
Stable Pairs

Abstract In 2008, Klemm–Pandharipande defined Gopakumar–Vafa type invariants of a Calabi–Yau 4-folds $X$ using Gromov–Witten theory. Recently, Cao–Maulik–Toda proposed a conjectural description of these invariants in terms of stable pair theory. When $X$ is the total space of the sum of two line bundles over a surface $S$, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on $S$. As an application, we obtain new verifications of the Cao–Maulik–Toda conjectures for low-degree curve classes and find connections to Carlsson–Okounkov numbers. Some of our verifications involve genus zero Gopakumar–Vafa type invariants recently determined in the context of the log-local principle by Bousseau–Brini–van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao–Maulik–Toda conjectures when thickened curves contribute and also for the case of local $\mathbb{P}^3$.

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.

Pseudo-Anosov Theory

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Braid Groups ◽  
Intersection Numbers ◽  
Branched Covers ◽  
Algebraic Integers ◽  
Dehn Twists ◽  
Mapping Classes ◽  
Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

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).

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