The sum of the Betti numbers of smooth Hilbert schemes

AbstractWe show that the Hilbert scheme, that parameterizes all ideals with the same Hilbert function over a Clements–Lindström ring W, is connected. More precisely, we prove that every graded ideal is connected by a sequence of deformations to the lex-plus-powers ideal with the same Hilbert function. This is an analogue of Hartshorne’s theorem that Grothendieck’s Hilbert scheme is connected. We also prove a conjecture by Gasharov, Hibi, and Peeva that the lex ideal attains maximal Betti numbers among all graded ideals in W with a fixed Hilbert function.

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Computation of the Betti numbers of Hilbert schemes

Hilbert Schemes of Zero-Dimensional Subschemes of Smooth Varieties - Lecture Notes in Mathematics ◽

10.1007/bfb0073493 ◽

1994 ◽

pp. 12-80

Author(s):

Lothar Göttsche

Keyword(s):

Betti Numbers ◽

Hilbert Schemes

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

The Electronic Journal of Combinatorics ◽

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.

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Extremal Betti numbers of graded ideals

Results in Mathematics ◽

10.1007/bf03322739 ◽

2003 ◽

Vol 43 (3-4) ◽

pp. 235-244 ◽

Cited By ~ 8

Author(s):

Marilena Crupi ◽

Rosanna Utano

Keyword(s):

Betti Numbers

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Curved Rickard complexes and link homologies

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2019-0044 ◽

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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The close relation between border and Pommaret marked bases

Collectanea mathematica ◽

10.1007/s13348-021-00313-w ◽

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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Derived Hilbert schemes

Journal of the American Mathematical Society ◽

10.1090/s0894-0347-02-00399-5 ◽

2002 ◽

Vol 15 (4) ◽

pp. 787-815 ◽

Cited By ~ 15

Author(s):

Ionuţ Ciocan-Fontanine ◽

Mikhail M. Kapranov

Keyword(s):

Hilbert Schemes

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Transverse Hilbert schemes and completely integrable systems

Complex Manifolds ◽

10.1515/coma-2017-0015 ◽

2017 ◽

Vol 4 (1) ◽

pp. 263-272 ◽

Cited By ~ 1

Author(s):

Niccolò Lora Lamia Donin

Keyword(s):

Integrable Systems ◽

Hilbert Scheme ◽

Symplectic Form ◽

Initial Surface ◽

Hilbert Schemes ◽

Completely Integrable Systems ◽

Completely Integrable ◽

Symplectic Surface ◽

Complex Symplectic ◽

Minimum Polynomial

Abstract In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].

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