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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 ([email protected])
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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 

References

1.Andreotti, A. and Norguet, F., La convexité holomorphe dans l'espace analytique des cycles d'une variété algébrique, Annali Scuola Norm. Sup. Pisa 21 (1967), 31–82.Google Scholar
2.Catanese, F., Chow varieties, Hilbert schemes and moduli spaces of surfaces of general type. J. Alg. Geom. 1(4) (1992), 561–595.Google Scholar
3.Cayley, A., On a new analytical representation of curves in space, Q. J. Pure Appl. Math. 3 (1860), 225–236.Google Scholar
4.Cayley, A., On a new analytical representation of curves in space, Q. J. Pure Appl. Math. 4 (1862), 81–86.Google Scholar
5.Dolgachev, I., Classical algebraic geometry (Cambridge University Press, 2012).Google Scholar
6.Ein, L., Varieties with small dual varieties, II, Duke Math. J. 52(4) (1985), 895–907.CrossRefGoogle Scholar
7.Ein, L., Varieties with small dual varieties, I, Invent. Math. 86(1) (1986), 63–74.CrossRefGoogle Scholar
8.Gelfand, I. M., Kapranov, M. M. and Zelevinski, A. V., Discriminants, resultants, and multidimensional determinants, Mathematics: Theory and Applications (Birkhäuser, 1994).CrossRefGoogle Scholar
9.Green, M. L. and Morrison, I., The equations defining Chow varieties, Duke Math. J. 53(3) (1986), 733–747.Google Scholar
10.Jessop, C. M., A treatise on the line complex (Chelsea, New York, 1969).Google Scholar
11.Mumford, D., Some footnotes to the work of C. P. Ramanujam, in C. P. Ramanujam–a tribute, Studies in Mathematics, Volume 8, pp. 247–262 (Springer, 1978).Google Scholar
12.Piene, R., Numerical characters of a curve in projective n-space, in Real and complex singularities: proceedings of the Nordic summer school/NAVF symposium in mathematics, Oslo, pp. 475–495 (Sijthoff and Noordhoff, Alphen aan den Rijn, 1977).Google Scholar
13.Popov, V. L., Self-dual algebraic varieties and nilpotent orbits, in Algebra, arithmetic and geometry, Part I, II: papers from the international colloquium held in Mumbai, January 4–12, 2000, Tata Institute of Fundamental Research Studies in Mathematics, Volume 16, pp. 509–533 (Narosa, Delhi, 2002).Google Scholar
14.Popov, V. L. and Tevelev, E. A., Self-dual projective algebraic varieties associated with symmetric spaces, in Algebraic transformation groups and algebraic varieties, Encyclopaedia of Mathematical Sciences, Volume 132, pp. 131–167 (Springer, 2004).Google Scholar
15.van der Waerden, B. L., Einführung in die algebraische Geometrie, Die Grundlehren der Mathematischen Wissenschaften, Band 51 (Springer, 1939).Google Scholar