The singularities of the secant curve associated to a space curve
Author:
Trygve Johnsen
Journal:
Trans. Amer. Math. Soc. 295 (1986), 107118
MSC:
Primary 14H45; Secondary 14H50, 14M15
MathSciNet review:
831191
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Abstract: Let be a curve in over an algebraically closed field of characteristic zero. We assume that is nonsingular and contains no plane component except possibly an irreducible conic. In [ ] one defines closed secant varieties to , . These varieties are embedded in , the Grassmannian of lines in . Denote by the secant variety (curve), and assume that the set of secants is finite. Let be the curve obtained by blowing up the ideal of secants in . The curve is in general not in . We study the local geometry of at any point whose fibre of the blowingup map is reduced at the point. The multiplicity of at such a point is determined in terms of the local geometry of at certain chosen secant points. Furthermore we give a geometrical interpretation of the tangential directions of at a singular point. We also give a criterion for whether all the tangential directions are distinct or not.
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 A. Andreotti, On a theorem of Torelli, Amer. J. Math. 80 (1958), 801828. MR 0102518 (21:1309)
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 L. Gruson and C. Peskine, Courbes de l'espace projectif, variétés de sécantes, Enumerative Geometry and Classical Algebraic Geometry, Progress in Math., Vol. 24, Birkhäuser, Basel, 1982.
 [J]
 T. Johnsen, A classification of the singularities of the secant curve associated to a space curve, Dr. Sci.Thesis, University of Oslo, 1984.
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 O. A. Laudal, Formal moduli of algebraic structures, Lecture Notes in Math., vol. 754, SpringerVerlag, 1979. MR 551624 (82h:14009)
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 , A generalized trisecant lemma, Algebraic Geometry, Proceedings, Tromsø, Norway, 1977, Lecture Notes in Math., vol. 687, SpringerVerlag, 1978, pp. 112149.
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 D. Mumford, Algebraic geometry I. Complex projective varieties, Grundlehren der Math. Wiss., vol. 221, SpringerVerlag, Berlin, 1976. MR 0453732 (56:11992)
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 P. Samuel, Lectures on old and new results on algebraic curves, Tata Inst. of Fundamental Research, Bombay, 1966. MR 0222088 (36:5140)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198608311913
PII:
S 00029947(1986)08311913
Keywords:
Space curve,
secant curve,
local geometry
Article copyright:
© Copyright 1986
American Mathematical Society
