Elementary components of Hilbert schemes

Abstract

We generalize the Bialynicki-Birula decomposition to singular schemes and apply it to the Hilbert scheme of points on an affine space. We find an infinite family of small, elementary and generically smooth components of the Hilbert scheme of points of the affine four-space. Our method gives easily verifiable sufficient conditions for proving that a point of the Hilbert scheme is smooth and lies on an elementary component. We also present a necessary condition for smoothability of a finite subscheme given by a homogeneous ideal.

post-final version, corrected proof of Thm 4.9

Cited by in corpus (5)

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• New elementary components of the Gorenstein locus of the Hilbert scheme of points
• Rational singularities of nested Hilbert schemes
• The motive of the Hilbert scheme of points in all dimensions
• Invariants of nested Hilbert and Quot schemes on surfaces