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Some manifolds generated by normal rational curves
Published online by Cambridge University Press: 24 October 2008
Extract
In a paper ‘The projective generation of curves and surfaces in space of four dimensions’, F. P. White considers the representation of prime sections of the general Bordiga surface F6 in [4] by quartic curves in the plane through ten fundamental points. We shall make use of several results from the same section of that paper, and, for convenience, list them as follows. From a point P′ of F6 can be drawn an ∞1 of trisecant lines lying on a cubic conical sheet. The curve of intersection of these trisecants and F6 is a curve C8, of order eight and with a double point at P′, which meets each of the ten lines on F6 in two points. In the plane, the representation of this curve is a curve C7, of order seven and with eleven double points, namely, the ten fundamental points and the point P which corresponds to P′. C7 is hyperelliptic, containing an involution of points Q, R corresponding to the points Q′, R′ in which a trisecant from P′ meets F6. The envelope of joints of corresponding points Q, R in the plane is a conic; this conic will continually be referred to as Σ.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 39 , Issue 3 , October 1943 , pp. 153 - 159
- Copyright
- Copyright © Cambridge Philosophical Society 1943
References
* Proc. Cambridge Phil. Soc. 21 (1922), 216–27.Google Scholar
* See Room, T. G., The Geometry of Determinantal Loci (Cambridge, 1938), p. 395.Google Scholar
* It is of interest that the plane representation of a double curve on a surface is, in this case, by two separate curves and not, as more usually, by a curve which has on it an involution 
. Usually the points forming coincident sets of such a correspond to pinch-points of the double curve lying on the surface; since each point of the double cubic on 
is represented in the plane by two points, P 1 on t 1 and P 2 on t 2, such that P 1P 2 is a tangent to ∑, and since P 1 and P 2 can never coincide, we see that the double curve on has no pinch-points. At every point of the double curve a general plane section of the surface has in fact only a node and not a cusp.
* Math. Ann. 3 (1870), 161.Google Scholar
† Acta Math. 6 (1936), 276.Google Scholar
‡ Principles of Geometry, 3 (Cambridge, 1934), 217.Google Scholar
* I.e. “chordal.”
† This result is very similar to a property of the general Bordiga sextic surface in [4] given by Room, loc. cit. p. 382; any trisecant plane to one of the ∞2 normal rational quartic curves on the Bordiga surface cuts it residually in three points on one of its ∞3 trisecant lines, and any plane through a trisecant line is trisecant to one of the quartic curves.
‡ Quart. J. of Math. 14 (1943), 5–15.Google Scholar