Degenerations of Complete Twisted Cubics

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


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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Copyright information

© Birkhäuser Boston, Inc. 1982

Authors and Affiliations

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

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