The singularities of the -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

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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 blowing-up 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0831191-3

Keywords:
Space curve,
-secant curve,
local geometry

Article copyright:
© Copyright 1986
American Mathematical Society