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Space sextic curves with six bitangents, and some geometry of the diagonal cubic surface

Published online by Cambridge University Press:  20 January 2009

R. H. Dye
R. H. Dye
Affiliation:
Department of Mathematics, University of Newcastle, Newcastle Upon Tyne NE1 7RU, England
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Abstract

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A general space curve has only a finite number of quadrisecants, and it is rare for these to be bitangents. We show that there are irreducible rational space sextics whose six quadrisecants are all bitangents. All such sextics are projectively equivalent, and they lie by pairs on diagonal cubic surfaces. The bitangents of such a related pair are the halves of the distinguished double-six of the diagonal cubic surface. No space sextic curve has more than six bitangents, and the only other types with six bitangents are certain (4,2) curves on quadrics. In the course of the argument we see that space sextics with at least six quadrisecants are either (4,2) or (5,1) quadric curves with infinitely many, or are curves which each lie on a unique, and non-singular, cubic surface and have one half of a double-six for quadrisecants.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 1 , February 1997 , pp. 85 - 97
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

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