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XII.—Some Remarks occasioned by the Geometry of the Veronese Surface

Published online by Cambridge University Press:  14 February 2012

W. L. Edge
W. L. Edge
Affiliation:
Mathematical Institute, University of Edinburgh
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Extract

The subject-matter of these pages may be briefly summarised as follows: the geometry of the Veronese surface, with an algebraic representation of it that does justice to its self-dual character; the relations of the secant planes of the surface to quadrics which either contain the surface or are outpolar to it; and the derivation of an invariant and two contravariants of a ternary quartic in the light of the (1, 1) correspondence between the quartic curves in a plane and the quadrics outpolar to a Veronese surface. There is no suggestion of discovering fresh properties of the surface, though possibly the results in § 12 § 13 may be new; but the geometrical considerations lead naturally to some algebraical results which it seems worth while to have on record, such as, for example, the identity 8.2 and the remarks concerning the rank of the determinant which appears there, and the form found in § 13 for the harmonic envelope of a plane quartic curve. These algebraical results lie very close to properties of the surface; so close in fact that one might say that the Veronese surface is the proper mise en scène for them.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 61 , Issue 2 , 1942 , pp. 140 - 159
Copyright
Copyright © Royal Society of Edinburgh 1942

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References

References to Literature

Bertini, E., 1923. Introduziont alia geotnetria proiettiva degli iperspazi, Seconda Edizione, Messina.Google Scholar
Cayley, A., 1856. “Note upon a result of elimination,” Phil. Mag., vol. xi, pp. 378–379; Collected Mathematical Papers, vol. iii, pp. 214–215.Google Scholar
Clebsch, A., 1861. “Über Curven vierter Ordnung,” Journ. reine angew. Math., vol. lix, pp. 125–145 (139).Google Scholar
Edge, W. L., 1938. “Notes on a net of quadric surfaces (III),” Proc. London Math. Soc. (2), vol. xliv, pp. 466–480.CrossRefGoogle Scholar
Muir, T., 1906 and 1911. The theory of determinants in the historical order of development, vol. i (London, 1906), p. 244; vol. ii (London, 1911), p. 52.Google Scholar
Pasch, M., 1891. “Über bilineare Formen und deren geometrische Anwendung,” Math. Ann., vol. xxxviii, pp. 24–49.Google Scholar
Scherrer, F. R., 1882. “Über ternäre biquadratische Formen,” Ann. Mat. pura appl. (2), vol. x, pp. 212–223.Google Scholar
Segre, C, 1885. “Considerazioni intorno alia geometria delle coniche di un piano e alia sua rappresentazione sulla geometria dei complessi lineari di rette,” Atti Ace. Torino, vol. xx, pp. 487–504.Google Scholar
Segre, C, 1892. “Alcune idee di Ettore Caporali intorno alle quartiche piane,” Ann. Mat. pura appl. (2), vol. xx, pp. 237–242.Google Scholar
Sylvester, J. J., 1841. “Examples of the dialytic method of elimination as applied to ternary systems of equations,” Cambridge Math. Journ., vol. ii, pp. 232–236; Collected Mathematical Papers, vol. i, pp. 61-65.Google Scholar
Veronese, G., 1884. “La superficie omaloide normale a due dimensioni e del quarto ordine dello spazio a cinque dimensioni e le sue proiezioni nel piano e nello spazio ordinario,” Mem. Accad. Lincei (3), vol. xix, pp. 344–371.Google Scholar