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# Degenerations of Complete Twisted Cubics

• Ragni Piene
Chapter
Part of the Progress in Mathematics book series (PM, volume 24)

## Abstract

Let C ⊂ P3 be a twisted cubic curve. Denote by Г ⊂ Grass(1,3) its tangent curve (curve of tangent lines) and by $${\text{C *}} \subset {\text{ }}\mathop {{{\text{P}}^3}}\limits^{\text{V}}$$ its dual curve (curve of osculating planes). The curve Г is rational normal, of degree 4, while C* is again a twisted cubic. The triple (C, Г, C*) is called a (non degenerate) complete twisted cubic. By a degeneration of it we mean a triple $${\text{(}}\overline {\text{C}} ,{\text{ }}\overline {\Gamma ,} {\text{ }}\overline {\text{C}} {\text{ *),}}$$ where $$\overline {\text{C}} ,{\text{ }}\overline \Gamma ,{\text{ }}\overline {\text{C}} *{\text{ }}\overline {\text{a}} {\text{re}}$$ simultaneous flat specializations of C, Г, C*. Thus we work with Hilbert schemes rather than Chow schemes: let H denote the irreducible component of Hilb3m+1 (P3) containing the twisted cubics, $$\mathop {\text{H}}\limits^{\text{V}}$$ the corresponding component of Hilb3m+1 $$(\mathop {{{\text{P}}^3}}\limits^{\text{V}} ),$$ and G the component of Hilb4m+1 (Grass(1, 3)) containing the tangent curves of twisted cubics. The space of complete twisted cubics is the closure T ⊂ H x G x $$\mathop {\text{H}}\limits^{\text{V}}$$ of the set of non degenerate complete twisted cubics.

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## Bibliography

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A.R. Alguneid, “Analytical degeneration of complete twisted cubics”, Proc. Cambridge Phil.Soc. 52 (1956), 202–208.
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R. Hartshorne, Algebraic Geometry. New York-Heidelberg -Berlin, Springer-Verlag 1977.
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## Copyright information

© Birkhäuser Boston, Inc. 1982

## Authors and Affiliations

• Ragni Piene
• 1
1. 1.Matematisk instituttBlindern, Oslo 3Norway