Varieties with degenerate dual variety

R. Muñoz

doi:10.1515/form.2001.033

Varieties with low dimensional dual variety

manuscripta mathematica ◽

10.1007/bf02677864 ◽

1997 ◽

Vol 94 (1) ◽

pp. 427-435 ◽

Cited By ~ 2

Author(s):

Roberto Muñoz

Keyword(s):

Dual Variety ◽

Low Dimensional

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DUAL VARIETY ROWS INCREASE ALMOND YIELDS

Acta Horticulturae ◽

10.17660/actahortic.1998.470.65 ◽

1998 ◽

pp. 469-472

Author(s):

John P. Edstrom ◽

Warren C. Micke

Keyword(s):

Dual Variety

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On the capacity of the circular disc condenser at small separation

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100001274 ◽

1970 ◽

Vol 68 (1) ◽

pp. 235-254 ◽

Cited By ~ 22

Author(s):

Frank Leppington ◽

Harold Levine

Keyword(s):

Integral Equation ◽

Asymptotic Expansion ◽

Boundary Value Problem ◽

Charge Density ◽

Boundary Value ◽

Dual Variety ◽

Long Span ◽

Infinite Extent ◽

Crude Estimate

AbstractA pair of identical circular discs, held at equal and opposite potentials, forms a condenser whose capacityCdepends on the ratio ε of separation against diameter. The determination of an asymptotic expansion forCwhen ε is small poses an axisymmetric boundary-value problem for harmonic functions that has engaged the attention of numerous investigators over a long span of time. It is a simple matter to construct a Fredholm integral equation of the first kind for the charge density ± σ on the discs, in terms of which the potential field and the capacity are implicitly determined, but the equation is unsuitable if ε ≪ 1. Integral equations of the second kind and of the dual variety have also been proposed as a means of securing a more manageable formulation of the boundary-value problem. An elementary approximation follows from the hypothesis that the charge density is almost the same as though the discs were of infinite extent, except for a region close to the edges, and leads to the resultC∼ l/8ε as ε → 0. Kirchhoff considerably improved on this crude estimate by suggesting a plausible edge correction which yields two further terms forC, of orders log ∈ and a constant, respectively, and his results have been rigorously established by the more refined analysis of Hutson. In the present work an integral equation of the first kind for the distribution of potential off the discs is derived and utilized to obtain an approximation forCwhen ε is small, reproducing the result of Kirchhoff and Hutson. Furthermore, an estimate of the error provides explicit details regarding the next term in the asymptotic expansion ofC, which is of the order ε(log ε)2.

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Cayley Forms and Self-Dual Varieties

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091513000928 ◽

2013 ◽

Vol 57 (1) ◽

pp. 89-109 ◽

Cited By ~ 3

Author(s):

F. Catanese

Keyword(s):

Quadratic Equation ◽

Space Curve ◽

Zero Set ◽

Dual Variety ◽

Quadratic Equations ◽

Dual Varieties ◽

Chow Forms ◽

The Ideal

AbstractGeneralized 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.

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