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I came across this notice, which seems to say Fano proved that a general cubic threefold is irrational back in 1940s. I'm interested in seeing this work, especially a proof without intermediate Jacobian. However, I did not find the original paper or a good exposition. Does anyone know Fano's approach?

Update: Carlo found Fano's original paper, which is great! However, I can hardly follow the mathematics in it (translated by machine) and I'm not sure if it is the language barrier. Meanwhile, I found a survey paper that cited Fano's paper saying Fano's work is not considered rigorous. So the question remains: What is Fano's method, and why it doesn't work?

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  • $\begingroup$ Indeed, I have added a link to a critical discussion of Fano's work. $\endgroup$
    – Carlo Beenakker
    Commented Dec 18, 2023 at 7:33
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    $\begingroup$ You can also find a critical discussion of Fano's work in Roth' Algebraic Threefolds (Springer Ergebnisse, 1955), around p. 94-96. Beware that this is not written in EGA style... $\endgroup$
    – abx
    Commented Dec 18, 2023 at 10:10
  • $\begingroup$ Dear @abx, thank you for the reference. This is probably the reference I'm looking for, but I do find it difficult to follow the exposition though, especially with unfamiliar terminologies and notations. Do you know if Fano's method was carried out later by someone else? $\endgroup$
    – AG learner
    Commented Dec 21, 2023 at 19:12
  • $\begingroup$ Yes, by Iskovskih and Manin in the paper quoted by Carlo. $\endgroup$
    – abx
    Commented Dec 22, 2023 at 5:34
  • $\begingroup$ Dear @abx, I’m particularly interested in cubic threefold, does birational rigidity proves irrationality of cubic threefold as well? $\endgroup$
    – AG learner
    Commented Dec 22, 2023 at 13:35

1 Answer 1

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The reference is Gino Fano, Nuove ricerche sulle varietà algebriche a tre dimensioni a curve-sezioni canoniche, published by the Papal Academy in 1948 (summarized in 1945):

It's in Italian (with an abstract in Latin); I guess Google translate can help.

Fano's proof is critically discussed by Iskovskih and Manin (1971):

The idea of the proof consists of verifying the incompatibility of singularities of large multiplicity $\nu_i$ on surfaces of small degree $4n$ with the unfixedness (or only positively) of this system of surfaces. In this connection it is necessary to examine a series of cases separately, depending on the position of the singularity $B_{i-1}$, with maximum multiplicity $\nu_i$. In almost all of these cases Fano's arguments do not withstand critical examination.

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