Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T15:37:17.654Z Has data issue: false hasContentIssue false
  • English
  • Français

The Cremona transformation of S6 by quadrics through a normal elliptic septimic scroll 1R7

Published online by Cambridge University Press:  26 February 2010

J. G. Semple  and
J. A. Tyrrell
J. G. Semple
Affiliation:
Mathematics Department, King's College London, Strand, W.C.2.
J. A. Tyrrell
Affiliation:
Mathematics Department, King's College London, Strand, W.C.2.
Get access

Extract

It is well known [1; pp. 333–339, 2; pp. 181–183] that the quadrics of S4 through a normal elliptic quintic curve 1C5 form a homaloidal system, and that this gives rise to an interesting Cremona transformation T2, 3 of S4 into S4′ for which the reverse system consists of the (Segre) cubic primals of S4′ that contain an elliptic quintic scroll 1R5. Less well known, though the fact was noted by one of us many years ago, is that the quadrics of S6 through the most general type of normal elliptic septimic scroll 1R7 likewise form a homaloidal system giving rise to a T2, 4 of S6 into S6′. We show in this paper that the latter transformation has properties no less remarkable than those of the above mentioned T2, 3. We propose also to show in a subsequent note that T2, 3 and T2, 4 are the initial members of a sequence of Cremona transformations T2, k+2 of S2k+2 (k = 1,2, …) of which the general member is generated by the homaloidal system of quadrics of S2k+2 that contain an elliptic ruled variety 1Rk2k+3, locus of an elliptic pencil of [k − 1]'s. Since the T2, 4 of S6, however, has certain cyclic properties peculiar to itself, we have thought it best to describe it separately in the present paper.

Type
Research Article
Information
Mathematika , Volume 16 , Issue 1 , June 1969 , pp. 88 - 97
Copyright
Copyright © University College London 1969

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

1.Semple, J. G., “Cremona transformations of space of four dimensions by means of quadrics, and the reverse transformations”, Phil. Trans. Royal Soc. of London Ser. A, 228 (1929), 331–376.Google Scholar
2.Semple, J. G. and Roth, L., Introduction to algebraic geometry (Oxford, 1949).Google Scholar
3.Enriques, F. and Chisini, O., Teoria geometrica delle equazioni e delle funzioni algebriche (Zanichelli, Bologna, Vol. III 1924, Vol. IV 1934).Google Scholar
4.Du Val, P., “Some rational curves representing modular relations”, Annali di mat. pura ed applicata Ser IV, 70 (1965), 249–270.CrossRefGoogle Scholar
5.Segre, C., “Remarques sur les transformations uniformes des courbes elliptiques en elles-mêmes”, Opere Vol. I, 36–55 (Rome, 1957).Google Scholar
6.Segre, C., “Ricerche sulle rigate ellittiche di qualunque ordine”, Opere Vol. I, 56–77 (Rome, 1957).Google Scholar
7.Klein, F., “Über die elliptischen Normalkurven der n-ten Ordnung”, Ges. Math. Behandlungen Vol. III, 198–254 (Berlin, 1923).Google Scholar
8.Edge, W. L., The theory of ruled surfaces (Cambridge, 1931).Google Scholar
9.Severi, F., “Sulle intersezioni delle varieta algebriche e sopra i loro caratteri e singolarità proiettive”, Memorie scelte, Vol. I, 41–115 (Bologna, 1950).Google Scholar
10.Semple, J. G. and Tyrrell, J. A., “Specialization of Cremona transformations”, Mathematika, 15 (1968), 171–177.CrossRefGoogle Scholar