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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
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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.

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Keywords: Space curve, $ 3$-secant curve, local geometry
Article copyright: © Copyright 1986 American Mathematical Society

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