Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T13:14:21.100Z Has data issue: false hasContentIssue false
  • English
  • Français

On quadrics through five real points

Published online by Cambridge University Press:  24 October 2008

R. H. F. Denniston
R. H. F. Denniston
Affiliation:
University of Leicester
Get access Rights & Permissions [Opens in a new window]

Extract

Let Q1,…, Q5 be five fixed points (no four coplanar) of the real projective space S3: let s be a variable quadric surface through these points. The set of all such quadrics can be represented by the points of a real S4, in which there is a quartic primal that represents cones. The geometry of this threefold is well known in the complex case, but has hardly been considered at all in the real case: and one object of the present paper is to describe the real threefold and determine its homology groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 2 , April 1967 , pp. 369 - 388
Copyright
Copyright © Cambridge Philosophical Society 1967

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Adamo, M.Alcune proprietà della di S 5 con sei punti doppî indipendenti, e sua trasformazione nella di C. Segre. Rend. Sem. Fac. Sci. Cagliari 27 (1957), 169–180.Google Scholar
(2)Baker, H. F.Principles of geometry, vol. 3 (Cambridge University Press, 1923).Google Scholar
(3)Baker, H. F.Principles of geometry, vol. 4 (Cambridge University Press, 1925).Google Scholar
(4)Castelnuovo, G.Ricerche di geometria della retta nello spazio a quattro dimensioni. Atti Ist. Veneto (7) 2 (38 of the complete series) (1891), 855–901.Google Scholar
(5)Dragoni, Angiola.Sulla varietà cubica di S 4 dotata di dieci punti doppî. Giorn. Mat. Battaglini 40 (1902), 255–264.Google Scholar
(6)Du Val, P.On questions of reality for twisted quartics of the first kind. Proc. Cambridge Phil. Soc. 24 (1928), 379–399.CrossRefGoogle Scholar
(7)Hudson, R. W. H. T.Kummer's quartic surface (Cambridge University Press, 1905).Google Scholar
(8)Klein, F.Über Flächen dritter Ordnung. Math. Ann. 6 (1873), 551–581.CrossRefGoogle Scholar
(9)Richmond, H. W.On the figure of six points in space of four dimensions. Quart. J. Pure Appl. Math. 31 (1900), 125–160.Google Scholar
(10)Richmond, H. W.Concerning the locus = 0; ∑(xr) = 0; (r = 1, 2, 3, 4, 5, 6). Quart. J. Pure Appl. Math. 34 (1903), 117–154.Google Scholar
(11)Segre, B.The non-singular cubic surfaces (Oxford, Clarendon Press, 1942).Google Scholar
(12)Todd, J. A.On questions of reality for certain geometrical loci. Proc. London Math. Soc. (2) 32 (1930), 449–487 (§6. The Segre cubic primal).Google Scholar