Abstract
REFERENCE has already been made in these columns (Nature, March 31, 1945, p. 385) to the proof by Prof. G. Fano, formerly of Turin, of a long-conjectured and important result in algebraic geometry. It has been known for nearly a century that the plane curve with equation f3 (x, y) = 0, where f3 denotes a cubic polynomial, is in general irrational, that is, its points cannot be set in one-to-one algebraic correspondence with the points of a line ; in fact, its co-ordinates are expressible as elliptic functions of a single parameter. In contrast to this the general cubic surface, given by an equation f3 (x, y,z) = 0, is rational, and the method of representing the surface biunivocally on a plane was obtained so long ago as 1866. It was thus natural to inquire whether the general cubic threefold, given by an equation f3 (x, y,z, t) = 0, is rational or not; the question posed by the Italian geometers some sixty years back long resisted all attempts at solution, and hence became one of the celebrated problems of geometry. In 1942, Fano (then in his seventieth year) at last established its irrationality ; from Switzerland, where he was living as an exile, he communicated the discovery to the Pontifical Academy of Sciences, and this, after various delays, has now been published with cognate researches ("Commentationes", 11, 635 ; 1947). Fano‘s proof, though conceptually simple, involves considerable detail, and, in fact, marks the conclusion to a long chain of investigations, dating from 1907, when Fano showed that the general quartic threefoldf4 (x, y,z, t) = 0 is irrational. It is interesting to note that the entire programme of the research was foreshadowed in a communication made to the International Mathematical Congress held at Bologna in 1928 ; this has now been carried out in all its details.
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Irrationality of the General Cubic Threefold. Nature 162, 563 (1948). https://doi.org/10.1038/162563b0
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DOI: https://doi.org/10.1038/162563b0