2019
DOI: 10.1112/jlms.12212
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Elementary components of Hilbert schemes of points

Abstract: Consider the Hilbert scheme of points on a higher dimensional affine space. Its component is elementary if it parameterizes irreducible subschemes. We characterize reduced elementary components in terms of tangent spaces and provide a computationally efficient way of finding such components. As an example, we find an infinite family of elementary and generically smooth components on the affine four-space. We analyse singularities and formulate a conjecture which would imply the non-reducedness of the Hilbert s… Show more

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Cited by 26 publications
(55 citation 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%
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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%
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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%
“…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]).…”
mentioning
confidence: 99%