A note on cotangent sheaves of Hilbert schemes of families of planar curves

We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.

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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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Affine Subspaces of Curvature Functions from Closed Planar Curves

Results in Mathematics ◽

10.1007/s00025-021-01378-6 ◽

2021 ◽

Vol 76 (2) ◽

Author(s):

Leonardo Alese

Keyword(s):

Arc Length ◽

Planar Curves ◽

Affine Subspaces ◽

Planar Curve ◽

Real Functions ◽

Discrete Counterpart ◽

Equivalent Conditions

AbstractGiven a pair of real functions (k, f), we study the conditions they must satisfy for $$k+\lambda f$$ k + λ f to be the curvature in the arc-length of a closed planar curve for all real $$\lambda $$ λ . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

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Clothoid fitting and geometric Hermite subdivision

Advances in Computational Mathematics ◽

10.1007/s10444-021-09876-5 ◽

2021 ◽

Vol 47 (4) ◽

Author(s):

Ulrich Reif ◽

Andreas Weinmann

Keyword(s):

Building Block ◽

Numerical Results ◽

Normal Vector ◽

Subdivision Schemes ◽

Planar Curves ◽

Nonlinear Subdivision ◽

New Strategy ◽

Vector Information ◽

Hermite Subdivision

AbstractWe consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.

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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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