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Cayley Forms and Self-Dual Varieties

Published online by Cambridge University Press:  19 December 2013

F. Catanese
F. Catanese*
Affiliation:
Lehrstuhl Mathematik VIII, Mathematisches Institut der Universität Bayreuth, NW II, Universitätsstrasse 30, 95447 Bayreuth (fabrizio.catanese@uni-bayreuth.de)
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Abstract

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Generalized Chow forms were introduced by Cayley for the case of 3-space; their zero set on the Grassmannian G(1, 3) is either the set Z of lines touching a given space curve (the case of an ‘honest’ Cayley form), or the set of lines tangent to a surface. Cayley gave some equations for F to be a generalized Cayley form, which should hold modulo the ideal generated by F and by the quadratic equation Q for G(1, 3). Our main result is that F is a Cayley form if and only if Z = G(1, 3) ∩ {F = 0} is equal to its dual variety. We also show that the variety of generalized Cayley forms is defined by quadratic equations, since there is a unique representative F0 + QF1 of F, with F0, F1 harmonic, such that the harmonic projection of the Cayley equation is identically 0. We also give new equations for honest Cayley forms, but show, with some calculations, that the variety of honest Cayley forms does not seem to be defined by quadratic and cubic equations.

Keywords

Cayley formsChow varietiesself-dual varietiesdevelopable surfacesharmonic representative polynomialsPrimary 14C05Secondary 14N9914M15
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 1 , February 2014 , pp. 89 - 109
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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