Białynicki-Birula decomposition for reductive groups

Jelisiejew, Joachim; Sienkiewicz, Łukasz

doi:10.48550/arxiv.1805.11558

Cited by 2 publications

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“…Upon completion of this work we learned that the Białynicki-Birula decomposition for algebraic spaces was constructed earlier by Drinfeld [Dri13], by an entirely different method. His method was extended [JS18] to actions of groups other than G m . While this paper was in review, a preprint [Jel18] appeared, which in particular claims to answer Question 1.5.…”

Section: Introductionmentioning

confidence: 99%

Elementary components of Hilbert schemes of points

Jelisiejew

2019

J. London Math. Soc.

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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 scheme. Our main tool is a generalization of the Bia lynicki-Birula decomposition for this singular scheme.

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Section: Introductionmentioning

confidence: 99%

Elementary components of Hilbert schemes of points

Jelisiejew

2019

J. London Math. Soc.

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“…Theorem 6.1. ([BB73, BB76], see also [Dri13,JS18]) Let X be a smooth projective scheme over an algebraically closed field k with a G m -action. We assume that X Gm is 0-dimensional.…”

Section: Smoothnessmentioning

confidence: 99%

“…The Bia lynicki-Birula's theorem gives us that any X + i is isomorphic to an affine space and {X + i } gives a cell decomposition of X [BB73, BB76]. Recently, for an arbitrary X locally of finite type, the BB schemes have been defined and investigated in [Dri13,JS18]. Thanks to [Dri13,JS18], we combine Bayer's degeneration and Bia lynicki-Birula's idea on the Hilbert scheme Hilb P n .…”

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Computable Białynicki-Birula decomposition of the Hilbert scheme

Kambe

2019

Preprint

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We call the scheme parameterizing homogeneous ideals with fixed initial ideal the Gröbner scheme. We introduce a Bia lynicki-Birula decomposition of the Hilbert scheme Hilb P n for any Hilbert polynomial P such that the cells are the Gröbner schemes in set-theoretically. Then we obtain a computable homology formula for smooth Hilbert schemes. As a corollary of our argument, we show that the Gröbner scheme for a monomial ideal defining a smooth point in the Hilbert scheme is smooth.

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Białynicki-Birula decomposition for reductive groups

Cited by 2 publications

References 0 publications

Elementary components of Hilbert schemes of points

Elementary components of Hilbert schemes of points

Computable Białynicki-Birula decomposition of the Hilbert scheme

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