Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The singularities of the $ 3$-secant curve associated to a space curve


Author: Trygve Johnsen
Journal: Trans. Amer. Math. Soc. 295 (1986), 107-118
MSC: Primary 14H45; Secondary 14H50, 14M15
MathSciNet review: 831191
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ C$ be a curve in $ {P^3}$ over an algebraically closed field of characteristic zero. We assume that $ C$ is nonsingular and contains no plane component except possibly an irreducible conic.

In [ $ {\mathbf{GP}}$] one defines closed $ r$-secant varieties to $ C$, $ r \in N$. These varieties are embedded in $ G$, the Grassmannian of lines in $ {P^3}$. Denote by $ T$ the $ 3$-secant variety (curve), and assume that the set of $ 4$-secants is finite. Let $ \tilde T$ be the curve obtained by blowing up the ideal of $ 4$-secants in $ T$. The curve $ \tilde T$ is in general not in $ G$.

We study the local geometry of $ \tilde T$ at any point whose fibre of the blowing-up map is reduced at the point. The multiplicity of $ \tilde T$ at such a point is determined in terms of the local geometry of $ C$ at certain chosen secant points. Furthermore we give a geometrical interpretation of the tangential directions of $ \tilde T$ at a singular point. We also give a criterion for whether all the tangential directions are distinct or not.


References [Enhancements On Off] (What's this?)

  • [Ab] Shreeram S. Abhyankar, Algebraic space curves, Les Presses de l’Université de Montréal, Montreal, Que., 1971. Séminaire de Mathématiques Supérieures, No. 43 (Été 1970). MR 0399109 (53 #2960)
  • [An] Aldo Andreotti, On a theorem of Torelli, Amer. J. Math. 80 (1958), 801–828. MR 0102518 (21 #1309)
  • [GP] 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 $ 3$-secant curve associated to a space curve, Dr. Sci.-Thesis, University of Oslo, 1984.
  • [La1] Olav Arnfinn Laudal, Formal moduli of algebraic structures, Lecture Notes in Mathematics, vol. 754, Springer, Berlin, 1979. MR 551624 (82h:14009)
  • [La2] -, A generalized tri-secant lemma, Algebraic Geometry, Proceedings, Tromsø, Norway, 1977, Lecture Notes in Math., vol. 687, Springer-Verlag, 1978, pp. 112-149.
  • [M] David Mumford, Algebraic geometry. I, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties; Grundlehren der Mathematischen Wissenschaften, No. 221. MR 0453732 (56 #11992)
  • [Sa] P. Samuel, Lectures on old and new results on algebraic curves, Notes by S. Anantharaman. Tata Institute of Fundamental Research Lectures on Mathematics, No. 36, Tata Institute of Fundamental Research, Bombay, 1966. MR 0222088 (36 #5140)
  • [WL] T. Wentzel-Larsen, Deformation theory of trisecant varieties, Preprint No. 1, Univ. of Oslo, 1985.
  • [ZS] Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581 (19,833e)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 14H45, 14H50, 14M15

Retrieve articles in all journals with MSC: 14H45, 14H50, 14M15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0831191-3
PII: S 0002-9947(1986)0831191-3
Keywords: Space curve, $ 3$-secant curve, local geometry
Article copyright: © Copyright 1986 American Mathematical Society