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On a class of unirational varieties

Published online by Cambridge University Press:  24 October 2008

L. Roth
L. Roth
Affiliation:
Imperial College of ScienceLondon, S.W. 7
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Extract

A theorem of Castelnuovo, which has played a considerable part in the general theory of surfaces, states that any surface which contains a net of elliptic curves is either rational or elliptic scrollar; more precisely, in the first case it is proved that the surface is unirational, and that its unirational representation is obtained by adjoining the irrationality on which depends the determination of one of its points, while the rest of the conclusion follows from Castelnuovo's theorem on the rationality of plane involutions. A somewhat similar result holds for surfaces which contain a net of hyperelliptic curves: thus, it is shown by Castelnuovo (loc. cit.) that, if the characteristic series of the net is not compounded of a g½ the surface is either rational or hyperelliptic scrollar.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 3 , July 1951 , pp. 496 - 503
Copyright
Copyright © Cambridge Philosophical Society 1951

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References

* Castelnuovo, G.Rend. Accad. Lincei, (5) 3 (1894)1, 473Google Scholar; Memorie scelte (Bologna, 1937), p. 233.Google Scholar

† A variety V r is unirational if it is representable on an involution I ν of S r; it is birational if ν = 1.

‡ Roth, L., Rend. Accad. Lincei, (8) 9 (1950), 62.Google Scholar

* A familiar example is the V 3 which is the complete intersection of a quadric and a cubic primal of [5]; this contains two systems of plane cubics, which can be separated only in an extension of the original field.

† The importance of this character is evident from Enriques's observation that, in general, it is impossible to lower the order of a given system of elliptic curves by means of rational processes.

‡ It follows from the concluding remark of § 1 that, since the curves C invade V r, the points Q must likewise do so; hence there cannot be fewer than ∞r−1 curves V 1, (P).

* Consider, for instance, a cubic primal of [4] and the system of plane sections which pass through a point P of the primal and which meet a fixed line. In this case the locus V 2(P) has a node at P.

* As in § 2, there cannot be fewer than ∞r−1 curves V 1.

* This method was suggested by an analogous device used in a paper by Segre, B.: Colloque international d'algèbre et de théorie des nombres (Paris, 1949), p. 135.Google Scholar

* Castelnuovo, G.-Enriques, F., Ann. Ecole Norm. Sup. (3), 22 (1906), 339.CrossRefGoogle Scholar