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A class of fundamental [n−1] Scrolls

Published online by Cambridge University Press:  24 October 2008

E. R. Woodcock
E. R. Woodcock
Affiliation:
Clare CollegeCambridge
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Extract

In a paper published in the Proceedings of the London Mathematical Society, (2), 40 (1936), 143, Welchman introduced the idea of fundamental scrolls from which all special scrolls may be obtained by projection, and he developed the theory of fundamental line scrolls. These fundamental line scrolls are generated by joins of pairs of corresponding points in a cyclic (1, 1) correspondence of period 2 on a canonical curve. In this paper I generalize some of Welchman's results, and then go on to consider certain fundamental scrolls which have a particular interest.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 36 , Issue 3 , July 1940 , pp. 288 - 299
Copyright
Copyright © Cambridge Philosophical Society 1940

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References

* Cf. Severi, , Trattato di geometria algebrica Bologna, 1926), p. 175.Google Scholar

† Defrise, P., Bull. Acad. Belgique Cl. Sci. (5), 24 (1938), 313Google Scholar and R.C. Semin. Mat. Univ. Roma, (4), 1 (1936), 83.Google Scholar

* Zeuthen's formula.

* Cf. Bertini, , Geometria proiettiva degli iperspazi (Messina, 1923), p. 80.Google Scholar

† Cf. Baker, , Principles of geometry, 6 (Cambridge 1933), 23.Google Scholar

‡ § 1.

§ Cf. Welchman, loc. cit. p. 148.

∥ Cf. Segre, C., R.C. Accad. Lincei, (4A), 3 (1887), 149Google Scholar. We can always find a proper n-secant curve of R(n) in [N] by taking a section by a general .

I.e. if i is the index of speciality of the prime sections of C.

* Since quadrics of this form are the only quadrics which transform into themselves under T′.

† Cf. Welchman, loc. cit. p. 151.

* Welchman, loc. cit. p. 156.

* There may of course be such a quadric even if this maximum condition is not satisfied.