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On the quintic surface in space of five dimensions

Published online by Cambridge University Press:  24 October 2008

F. Bath
F. Bath
Affiliation:
King's College
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Extract

Surfaces of order n in space of n dimensions, for 3 ≤ n ≤ 9, were discussed by Del Pezzo, who showed that the prime sections of such a surface are represented on the plane by cubic curves through (9 − n) base points.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 24 , Issue 1 , January 1928 , pp. 48 - 55
Copyright
Copyright © Cambridge Philosophical Society 1928

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References

* “Sulle superficie dell' n mo ordine immerse negli spazi di n dimensioni,” Rendiconti Circolo Matematico di Palermo, Tomo I (1887).Google Scholar

† “Sulle intersezioni delle varietà algebriche…,” Memorie Torino (2), 52 (1903) p. 102.Google Scholar

‡ “Surfaces on which lie Five Systems of Conics,” Proceedings London Mathematical Society, Vol. 24 (12 1924), p. v.Google Scholar

§ In this paper, unless otherwise stated, ‘quadric’ will mean ‘quadric fourfold’: the symbol Qn will be used to denote a quadric n-fold.Google Scholar

* Cf. Enriques-Chisini, , Teoria Geometrica delle Equazioni…, Vol. 3, p. 106;Google Scholar also Enriques, , “Sulle curve canoniche di genere p dello spazio a p − 1 dimeneioni,” Rendiconti Bologna, 23 (1919), p. 80 (not “Atti Bologna” as given in Vol. 3 above quoted).Google Scholar

† l ik = l ki.Google Scholar

* The configuration has other interesting properties: for example, there are 145 triads of the fifteen points, each triad containing no pair of points lying on one of the ten lines. These triads define planes which pass by fives through certain points.Google Scholar

* Writing , the quartics are

all of which pass through the point (1, 1, − 1 ), have double points at (1, 0, 0) and (0, 1, 0) and y = 0, x = 0 as tangents at (1, 0, 0), (0, 1, 0) respectively. Taking (1, 0, 0), (0, 1, 0) and (1, 1, − 1) as the triangle of reference and applying the transformation x′:y′:z′ = YZ : ZX : XY, the system is transformed into a system of cubics through four fixed points.

* Otherwise, the line l 12 is the locus of points (4.1) for which λ1λ2 = 1, or from the plane representation as shown later.Google Scholar

† These are the only points in common since the seven quadric twofolds are linearly independent.Google Scholar