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On the Hilbert scheme compactification of the space of twisted cubics. (English) Zbl 0589.14009
The main theorem of this paper is that the Hilbert scheme compactification of the space of twisted cubic curves is smooth. This is shown by an explicit computation of the universal deformation of the worst possible flat degeneration of a twisted cubic. Along the way the authors prove the following result, now commonly known as the Piene- Schlessinger comparison theorem: Suppose X in \({\mathbb{P}}^ n\) is defined by homogeneous polynomials \(f_ 1,...,f_ r\) of degrees \(d_ 1,...,d_ r\), respectively, and that the linear systems cut out by hypersurfaces of degrees \(d_ 1,...,d_ r\) on X are complete. Then any infinitesimal deformation of X is induced by a unique deformation of the affine cone over X.
Reviewer: S.A.Strømme

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
14D22 Fine and coarse moduli spaces
14D15 Formal methods and deformations in algebraic geometry
14H45 Special algebraic curves and curves of low genus
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