Elementary components of Hilbert schemes of points

Jelisiejew, Joachim

doi:10.1112/jlms.12212

Cited by 26 publications

(55 citation statements)

References 49 publications

(83 reference statements)

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“…The main aim of this paper is to generalize the results of [Dri13,Jel17] by replacing G m with an arbitrary linearly reductive affine group G. The functorial description (1) readily generalizes once we understand what should be put in place of A 1 . It turns out that a suitable replacement is a linearly reductive monoid G, i.e., an affine variety with multiplication G × G → G and a unit, such that G ⊂ G is the dense submonoid consisting of invertible elements.…”

Section: Introductionmentioning

confidence: 95%

“…The space X + defined in (1) is an important tool for analysis of highly singular spaces, such as moduli spaces. For example, X + and the morphism X + → X is used in [Jel17] to analyse X = Hilb d A n . In this setting, the space X + has a serious drawback: its definition takes into account only a fixed G m -action on X, while the automorphism group of X is usually larger (for Hilb d A n we have for example the action of GL n ).…”

Section: Introductionmentioning

confidence: 99%

“…Proposition 1.6 has the following consequence, which is of independent interest: if x lies in X G and there exists a linearly reductive monoid G with zero such that T X,x has no outsider representations (for G), then there exists a G-stable affine open neighbourhood of x. Proposition 1.6 is also motivated by the results of [Jel17], in particular by [Jel17, Proposition 1.5] and we plan to investigate its consequences for Hilbert and Quot schemes.…”

Section: Introductionmentioning

confidence: 99%

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Białynicki-Birula decomposition for reductive groups

Jelisiejew

Sienkiewicz

2019

Journal de Mathématiques Pures et Appliquées

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We generalize the Białynicki-Birula decomposition from actions of G m on smooth varieties to actions of linearly reductive group G on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks.We define the Białynicki-Birula decomposition functorially: for a fixed G-scheme X and a monoid G which partially compactifies G, the BB decomposition parameterizes G-schemes over X for which the G-action extends to the G-action. The freedom of choice of G makes the theory richer than the G m -case.Corollary 1.2. Let X be a G-scheme locally of finite type over k. Then every point of X has an affine G-stable neighbourhood.Proof. Note that X = X + by Proposition 5.7. By Theorem 1.1 the projection π X is G-equivariant and affine. The required neighbourhood of x ∈ X is the preimage of any affine open neighbourhood of π X (x) ∈ X G . This is striking, since for an arbitrary, even normal, G-scheme X the existence of affine G-stable neighbourhoods is subtle, see Brion [Bri15]. As an example of non-normal scheme: the node of the nodal curve C = P 1 /(0 = ∞) with its natural G m -action does not admit an affine G m -stable neighbourhood; here C + is equal to A 1 . Now we describe X + for an affine X. Recall that every irreducible finite dimensional Grepresentation appears in H 0

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Białynicki-Birula decomposition for reductive groups

Jelisiejew

Sienkiewicz

2019

Journal de Mathématiques Pures et Appliquées

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“…This observation motivates interest for the so-called elementary components of a punctual Hilbert scheme, i.e. components whose points parameterize zero-dimensional subschemes with support of cardinality one (see [Jelisiejew, 2017] for a very recent contribution in this context). Hence, the problem of detecting smoothable points is connected to the study of ideals I such that R/I is a local K-algebra.…”

Section: Introductionmentioning

confidence: 99%

Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals

Bertone

Cioffi

Roggero

2019

Experimental Mathematics

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Over an infinite field K with char(K) = 2, 3, we investigate smoothable Gorenstein Kpoints in a punctual Hilbert scheme from a new point of view, which is based on properties of double-generic initial ideals and of marked schemes. We obtain the following results: (i) points defined by graded Gorenstein K-algebras with Hilbert function (1, 7, 7, 1) are smoothable, in the further hypothesis that K is algebraically closed; (ii) the Hilbert scheme Hilb 7 16 has at least three irreducible components. The properties of marked schemes give us a simple method to compute the Zariski tangent space to a Hilbert scheme at a given K-point, which is very useful in this context. Over an algebraically closed field of characteristic 0, we also test our tools to find the already known result that points defined by graded Gorenstein K-algebras with Hilbert function (1, 5, 5, 1) are smoothable. In characteristic zero, all the results about smoothable points also hold for local Artin Gorenstein K-algebras.

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“…More is known about some particular Hilbert schemes or some special sub-loci. The case of punctual Hilbert schemes has been studied continuously since the 70s (see [25] and references therein), and it is still under investigation nowadays [8,24,27,28,37]. In the case of 1-dimensional subschemes of the projective space P 3 there is a remarkable variety of results (for instance about ACM curves, see [13,42,14,5]).…”

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confidence: 99%

The Gröbner fan of the Hilbert scheme

Kambe

Lella

2020

Annali di Matematica

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We give a notion of "combinatorial proximity" among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees "geometric proximity" of the corresponding points in the Hilbert scheme. We define a graph whose vertices correspond to strongly stable ideals and whose edges correspond to pairs of adjacent ideals. Every term order induces an orientation of the edges of the graph. This directed graph describes the behavior of the points of the Hilbert scheme under Gr öbner degenerations with respect to the given term order.Then, we introduce a polyhedral fan that we call Gröbner fan of the Hilbert scheme. Each cone of maximal dimension corresponds to a different directed graph induced by a term order. This fan encodes several properties of the Hilbert scheme. We use these tools to present a new proof of the connectedness of the Hilbert scheme. Finally, we improve the technique introduced in the paper "Double-generic initial ideal and Hilbert scheme" [5] (Bertone, Cioffi, Roggero, Ann. Mat. Pura Appl. ( 4) 196(1), 2017) to give a lower bound on the number of irreducible components of the Hilbert scheme.

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Elementary components of Hilbert schemes of points

Cited by 26 publications

References 49 publications

Białynicki-Birula decomposition for reductive groups

Białynicki-Birula decomposition for reductive groups

Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals

The Gröbner fan of the Hilbert scheme

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