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I am interested to know the list of non-rational smooth Fano 3-folds with Picard number greater than 1 (more precisely families where at least one smooth member is known to be non-rational). It seems that there are not so many.

In the list of Fano 3-folds that I have at hand, it states whether each Fano 3-fold is rational or not in almost all cases. But unfortunately, there is a ? written next to one of the varieties and the authors leave a gap next to 5 other cases.

I am not sure whether these omissions are due to the rationality of these varieties being an open problem, or whether it is due to some complicated situation with some rational and non-rational varieties in the family. Any enlightenment about the state of the art would be greatly appreciated.

The examples I did manage to obtain from this list are (in no particular order):

1.Double cover of $\mathbb{P}^1 \times \mathbb{P}^2$, branched along a divisor of bi-degree $(2,4)$.

  1. Blow up of $V_{1}$ (Del-pezzo 3-fold of degree 1) in the intersection of two divisors from the anti-canonical class.

  2. Blow-up of cubic 3-fold in a line.

  3. Blow-up of cubic 3-fold in a planar cubic curve.

  4. A divisor in $\mathbb{P}^2 \times \mathbb{P}^2$ of bi-degree (2,2).

  5. Double cover of $V_{7}$ ($= Bl_{p} \mathbb{P}^3$) branched in a divisor in the anti-canonical class (satisfying certain smoothness condition).

  6. Double cover of $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, branched along a smooth divisor of tri-degree (2,2,2).

(1-6. have $\rho(X) = 2$ and 7. has $\rho(X) = 3$.)

Question: Is there any 3-folds missing from the list? i.e. is there a smooth Fano 3-fold $X$ with $\rho(X)$>1, that is known to be non-rational that is not contained in one of these 7 families.

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  • $\begingroup$ You forgot 2 or 3 cases. You'll find a complete list at the end of Fano varieties by Iskovskikh and Prokhorov (Algebraic geometry, V, 1-247, Encyclopaedia Math. Sci., 47, Springer). $\endgroup$ – abx Jul 5 '17 at 16:26
  • $\begingroup$ Thanks, this is the list that I used originally. In the list at the end they use U to denote that this 3-fold is unirational but non-rational and R that is Rational. Above I have listed all the 3-folds with a U next to it. There are 5 3-fold with nothing in this box. One of these "blank cases" is a blow-up of the variety of full-flags in $\mathbb{C}^{3}$ in a curve hence rational, so I am very confused... (This is number 7 in the list with $\rho(X) = 3$). So as far as I can tell blank does not mean anything specific. I would be very grateful if you could indicate a specific case I have missed. $\endgroup$ – Nick L Jul 5 '17 at 17:49
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    $\begingroup$ The blow-up of $V_2$ (nr. 3 with $\rho =2$), the double cover of $W$ (nr. 6 b) with $\rho =2$). Also, in your nr. 6, you want a double cover and not a blow-up. $\endgroup$ – abx Jul 5 '17 at 18:31
  • $\begingroup$ Thanks! It turns out that all the other blank cases (other than nr. 8 with $\rho = 3$) are curve blow-ups of 3-folds that are well known to be rational, I don't know why I didn't spot this! So modulo determining whether the case of number 8 in the $\rho =3$ is rational, I think this is the complete list. $\endgroup$ – Nick L Jul 5 '17 at 21:00
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    $\begingroup$ @NickL: Nr. 8 is a conic bundle over $P^1 \times P^1$, so one can compute its intermediate Jacobian. $\endgroup$ – Sasha Jul 7 '17 at 8:33
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A complete list of non-rational Fano threefolds, together with results towards non-stable rationality, can be found in the paper:

BRENDAN HASSETT AND YURI TSCHINKEL- ON STABLE RATIONALITY OF FANO THREEFOLDS AND DEL PEZZO FIBRATIONS

http://www.math.nyu.edu/~tschinke/papers/yuri/16fano3/fano3-9.pdf

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    $\begingroup$ Thanks! this is definitely a relevant paper. As far as I can tell, they only list minimal non-rational Fano 3-folds (I guess it means ones that are not curve blow-ups) which is why the list has only 4 entries at the end of section 2. This is a useful statement, but I am really interested to know all of the 3-folds with this property (which has at least 9 entries by the question + comments of abx). $\endgroup$ – Nick L Jul 5 '17 at 21:18
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I think you might be interested in the following article (in italian):

"Sulla razionalità delle 3-varietà di Fano con B2 almeno 2" by Alzati and Bertolini (Le Matematiche, Vol. XLVII (1992) - Fasc. I, pp. 63-74)

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