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The Enriques threefold

Published online by Cambridge University Press:  24 October 2008

J. A. Tyrrell
J. A. Tyrrell
Affiliation:
King's CollegeLondon
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Extract

It is a classical and well-established result that a unirational algebraic variety of dimension ≪ 2 is necessarily rational. It is also generally agreed, though not so well established, that this result is no longer true for varieties of dimension three. In this connexion, the critical example has been the Enriques threefold E, asserted by Roth (1) to be unirational but not rational. The purpose of this note is to point out a fallacy in the proof of the non-rationality of E and, incidentally, to resolve a difficulty raised by Serre (2) in which E would appear to feature as a counter-example to a well-established general theorem. As far as the author is aware, there are no other examples of nonrational unirational threefolds, so that the question of their existence is still open.

Type
Research Notes
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 4 , October 1961 , pp. 897 - 898
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

REFERENCES

(1)Roth, L., Algebraic threefolds, with special regard to problems of rationality (Berlin, 1955).Google Scholar
(2)Serre, J.-P., On the fundamental group of a unirational variety. J. Lond. Math. Soc. 34 (1959), 481–4.CrossRefGoogle Scholar