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A special net of quadrics

Published online by Cambridge University Press:  20 January 2009

W. L. Edge
W. L. Edge
Affiliation:
University of Edinburgh.
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It is well-known that a general net of quadric surfaces cannot be obtained as the net of polar quadrics of the points of a plane in regard to a cubic surface; in order that it may be so obtained it must have various properties that a general net of quadrics does not have. The locus of the vertices of the cones which belong to the net of quadrics is a curve ϑ –the Jacobian curve of the net of quadrics, and the trisecants of ϑ generate a scroll. Any plane which contains two trisecants of ϑ is a bitangent plane of the scroll and, for a general net of quadrics, there are eighteen of these bitangent planes passing through an arbitrary point. When however the net of quadrics is a net of polar quadrics it is found that any plane which contains two trisecants of ϑ contains two other trisecants also; it thus contains four trisecants in all and counts six times as a bitangent plane of the scroll. The bitangent developable of the scroll, which is, for a general net of quadrics, of class eighteen, degenerates, in the special case when the net of quadrics is a net of polar quadrics, into a developable of class three counted six times; the planes of the developable are therefore the osculating planes of a twisted cubic γ. The plane which, together with a cubic surface, gives rise to the net of polar quadrics must be one of the osculating planes of γ. It is also found, further, that the osculating planes of γ are grouped into pentahedra, the vertices of all these pentahedra lying on ϑ.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 4 , Issue 4 , April 1936 , pp. 185 - 209
Copyright
Copyright © Edinburgh Mathematical Society 1936

References

page 185 note 1 Math. Annalen, 18 (1881); see in particular pp. 23–27.Google Scholar See also Töplitz, : Math. Annalen, 11 (1877), 434–463;CrossRefGoogle Scholar Töplitz proves that the ∞1 planes which give rise to the net of quadrics all osculate the same twisted cubic. There is also a paper by Dixon, A.C.: Proa. London Math. Soc. (2), 7 (1909), 150–156, in which he gives an algebraical proof that the faces of the pentahedra all osculate the twisted cubic obtained by Töplitz, but he seems unaware of Schur's work.CrossRefGoogle Scholar

page 187 note 1 The equation of the doubly-infinite family of cubic surfaces contains the parameter λ to the first degree and the parameter a to degree 14. Töplitz states, in the footnote to p. 449 of his paper, that this second parameter enters to degree 8. I believe however that the word achten is a misprint for achtzehnten, since it seems, on reading carefully through his work, that the second parameter enters to degree 18 in his equation (9). He goes on to say that it does not seem possible to lessen this degree, but I have not been able to account for the discrepancy between his 18 and the degree 14 to which a enters iu the above equation. This equation, with a entering to degree 14, is also implied in Dixon's work on pp. 154-155 of his paper.

page 187 note 2 See, for example, Baker, , Principles of Geometry 3 (Cambridge, 1933), 206–208.Google Scholar

page 188 note 1 Acta mathematica 66 (1936), 253–332.CrossRefGoogle Scholar

page 190 note 1 Suppose we have a matrix of p rows and q columns (p≤q), the elements of the matrix being homogeneous polynomials in the coordinates of a point in [n], all those elements in the i th row being of degree r i. Then the locus given by the simultaneous vanishing of all the determinants of p rows and columns belonging to the matrix is of dimension n-(q – p + l), and its order is the coefficient of t q–p+1 in the expansion of {(1–r 1t)(1–r 2t)……(1–r pt)}–1. See Baker, : Principles of Geometry 6 (Cambridge, 1933), 109Google Scholar. Here p=5; r 1 =r 2 =0, r 3 =r 4 =r 5 =1.

page 194 note 1 If Γ0 =0, Γ1=0, Γ2 =0 are the equations of three independent quadrics of the net, the coordinates of such a point must cause all the three-rowed determinants of the matrix , of three rows and n + 1 columns, to vanish.

page 195 note 1 When n = 3 the forms of the equations of the three quadrics arising from these identities are equivalent to the less symmetrical set of three equations given by Dixon, , loc. c.it., p. 154.Google Scholar

page 199 note 1 Proc. Edinburgh Math. Soc. (2), 3 (1933), 263.Google Scholar

page 202 note 1 See, for example, Edge, : The Theory of Buled Surfaces (Cambridge, 1931), pp. 17–18;Google Scholar or Baker, , Principles of Geometry, 6 (Cambridge, 1933), p. 16.Google Scholar

page 205 note 1 Edge, : Proc. London Math. Soc. (2), 33 (1932), 53–54.Google Scholar

page 206 note 1 Zeuthen, : Lehrbuch der abzählenden Methoden der Geotnetrie (1914), 104–107;Google Scholar or Math. Annalen 3 (1871), 150.Google Scholar See also Baker, : Principles of Geometry 6 (Cambridge 1933), 19. The formula for an (a, a′) correspondence between two curves of genera p and p′ isGoogle Scholar

where η, η′ are the numbers of branch points of the correspondence on the two curves. The formula applies not merely to curves, but to any loci generated by singly-infinite sets of elements.