Algebraic Families on an Algebraic Surface, II, the Picard Scheme of the Punctual Hilbert Scheme

1973 ◽  
Vol 95 (3) ◽  
pp. 660 ◽  
Author(s):  
J. Fogarty
Keyword(s):  
Hilbert Scheme ◽  
Algebraic Surface ◽  
Punctual Hilbert Scheme ◽  
Picard Scheme

scholarly journals Dimension of the punctual Hilbert scheme

1978 ◽  
Vol 55 (2) ◽  
pp. 536-544 ◽  
Author(s):  
J Briançon ◽  
A Iarrobino
Keyword(s):  
Hilbert Scheme ◽  
Punctual Hilbert Scheme

scholarly journals On the Gorenstein locus of the punctual Hilbert scheme of degree 11

2014 ◽  
Vol 218 (9) ◽  
pp. 1635-1651 ◽  
Author(s):  
Gianfranco Casnati ◽  
Roberto Notari
Keyword(s):  
Hilbert Scheme ◽  
Punctual Hilbert Scheme

scholarly journals Local finite-dimensional Gorenstein k-algebras having Hilbert function (1,5,5,1) are smoothable

2014 ◽  
Vol 13 (08) ◽  
pp. 1450056 ◽  
Author(s):  
Joachim Jelisiejew
Keyword(s):  
Hilbert Scheme ◽  
Hilbert Function ◽  
Hilbert Schemes ◽  
Gorenstein Algebras ◽  
Punctual Hilbert Scheme ◽  
Finite Dimensional ◽  
Hilbert Scheme Of Points

We consider the question of irreducibility of the Hilbert scheme of points ℋilbdℙn and its Gorenstein locus. This locus is known to be reducible for d ≥ 14. For d ≤ 11 the irreducibility of this locus was proved in the series of papers [G. Casnati and R. Notari, On the Gorenstein locus of some punctual Hilbert schemes, J. Pure Appl. Algebra 213(11) (2009) 2055–2074; On the irreducibility and the singularities of Gorenstein locus of the Punctual Hilbert scheme of degree 10, J. Pure Appl. Algebra 215(6) (2011) 1243–1254; Irreducibility of the Gorenstein locus of the Punctual Hilbert Scheme of degree 11, preprint (2012)] and Iarrobino conjectured that the irreducibility holds for d ≤ 13. In this paper, we prove that the subschemes corresponding to the Gorenstein algebras having Hilbert function (1, 5, 5, 1) are smoothable, i.e. lie in the closure of the locus of smooth subschemes. This result completes the proof of irreducibility of the Gorenstein locus of ℋilb12ℙn, see Theorem 2.

scholarly journals The tangent space of the punctual Hilbert scheme

2017 ◽  
Vol 66 (3) ◽  
pp. 595-610 ◽  
Author(s):  
Dori Bejleri ◽  
David Stapleton
Keyword(s):  
Hilbert Scheme ◽  
Tangent Space ◽  
Punctual Hilbert Scheme

scholarly journals Generalized punctual Hilbert schemes and 𝔤-complex structures

2021 ◽  
Author(s):  
Alexander Thomas
Keyword(s):  
Lie Algebras ◽  
Moduli Spaces ◽  
Hilbert Scheme ◽  
Complex Structure ◽  
Complex Structures ◽  
Hilbert Schemes ◽  
Simple Complex ◽  
Geometric Structures ◽  
Punctual Hilbert Scheme ◽  
Alternative Description

We define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple properties with Hitchin components, and which are conjecturally homeomorphic to them. For simple complex Lie algebras, this generalizes the higher complex structure. For real Lie algebras, this should give an alternative description of the Hitchin–Kostant–Rallis section.

scholarly journals Higher Complex Structures

2019 ◽  
Author(s):  
Vladimir Fock ◽  
Alexander Thomas
Keyword(s):  
Moduli Space ◽  
Geometric Structure ◽  
Hilbert Scheme ◽  
Complex Structure ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Complex Structures ◽  
Punctual Hilbert Scheme

Abstract We introduce and analyze a new geometric structure on topological surfaces generalizing the complex structure. To define this so-called higher complex structure, we use the punctual Hilbert scheme of the plane. The moduli space of higher complex structures is defined and is shown to be a generalization of the classical Teichmüller space. We give arguments for the conjectural isomorphism between the moduli space of higher complex structures and Hitchin’s component.

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