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On Certain Loci of Three Dimensions Representable on Ordinary Space by means of Cubic Surfaces, and the Cremona Transformations for Ordinary Space obtained by Projection of such Loci

Published online by Cambridge University Press:  24 October 2008

J. G. Semple
J. G. Semple
Affiliation:
St John's College.
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Extract

If we have one birational representation of a threefold locus V on an ordinary space S3, and if we can project V birationally on to another space S3′, then clearly there will be a Cremona correspondence between the two spaces S3 and S3′. This paper deals, in the first place, with threefold loci in higher space which can be represented birationally on ordinary space by means of cubic surfaces; and Cremona transformations for ordinary space are then obtained, as indicated above, by projecting such loci. In particular it will be shown that most of the more important cubic transformations and their reverses can be obtained very simply by this method.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 25 , Issue 2 , April 1929 , pp. 145 - 167
Copyright
Copyright © Cambridge Philosophical Society 1929

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References

* When a locus is denoted by a letter with an upper and a lower index, say, the upper index will always be the order, and the lower index the dimension of the locus.Google Scholar

* We shall often use the symbol [n] to denote a space of n dimensions.Google Scholar

* The system of all quadrics in S3 is of freedom nine and grade eight; all such quadrics can therefore be represented on a of [9], analogous to the Veronese surface in [5], which represents all the conics of the plane. This and its projections are well known. (Cf. for example, Segre, , Ency. d. math. Wiss. III c 7, p 958, note 580.)Google Scholar

† We shall use the letters φ and φ′ throughout this paper to denote the direct and reverse homaloidal systems associated with a Cremona transformation.Google Scholar

* For the literature about the various Cremona cubic transformations, obtained in this paper by projection from higher space, we refer to Miss Hudson's Cremona Transformations (Cambridge, 1927). This contains, on pp. 447–8, an exhaustive list of homaloidal systems of cubic surfaces, and gives complete references to the relevant literature.Google Scholar

The above transformation is the third on Miss Hudson's list.Google Scholar

† Mem. Acc. Torino (2) 39, (1888), p. 3.Google Scholar

‡ A general line of S3 meets every surface of Φ0 in three free points, and so corresponds to a cubic curve on . But if the line contains two base points of Φ0, it will meet each surface of Φ0 in only one free point, and so will correspond to a line only of .Google Scholar

* It can be shown that three quadrics in [4] which pass through a rational cubic curve meet further in an elliptic quintic.Google Scholar

* Cf. Baker, , Principles of Geometry, Vol. IV, pp. 234–5.Google Scholar

* For n=4, for example, is the rational normal quintic surface, with prime sections of genus two, representable on the plane by quartics with one double and seven simple base points.Google Scholar

* By “linear” sections of a locus, we mean sections by linear spaces of such dimension that they meet the locus in curves.Google Scholar

† Math. Ann., 46 (1895), p. 179.CrossRefGoogle Scholar

‡ Ann. di. Math. (3) 15 (1908), p. 217.Google Scholar

§ Cf. del Pezzo, , Rend, del Circ. Mat. di Palermo, 1 (1887), p. 241.Google Scholar

* The section, in question is a surface of order five in [5] and this has five conics through any point of itself, as is easily seen from its plane representation.Google Scholar

† Conies which meet K2 project from K2 to secants of the rational quartic curve C4 in S3; and those which also touch σ at P project to the lines which join P′, the projection of P, to the four points in which the plane σ′, the projection of σ, meets C4; i.e. there are four of them.Google Scholar

‡ Scorza, , in his paper (already quoted) shows that for n>6 the locus in question is a cone.6+the+locus+in+question+is+a+cone.>Google Scholar

* For the surface corresponding to any plane σ of S3 is represented on σ by means of cubic curves through the four points in which σ is met by C4.Google Scholar

* Any linear section of residual to l, will clearly meet l in three points; for two cubic surfaces in S3 which pass through C4 and have a common node D on C4 must meet in a residual quintic curve with a triple point at D. This shows that projects from l unto a of [5].Google Scholar

† For all chords of C4, through a given point of the curve, form a surface of Φ, an elliptic cubic cone, in fact.Google Scholar

* This is another of the loci with elliptic linear sections obtained by Enriques in his paper already quoted.Google Scholar

† By a “secundum” we mean a space of dimension two less than that in which we are working.Google Scholar