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Does there exist a smooth non-rational projective variety whose bounded derived category of coherent sheaves admits a full exceptional collection? I could not find any examples in the literature (for instance projective spaces and intersections of quadrics, which admit full exceptional collections, are rational).

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    $\begingroup$ Perhaps this question should count as an "open problem". $\endgroup$ – Leo Alonso Aug 31 '18 at 14:28
  • $\begingroup$ @LeoAlonso is it? I did not know it. If you could post a link to a paper where it is stated to be an open problem, I would accept it as an answer. $\endgroup$ – user74900 Aug 31 '18 at 14:36
  • $\begingroup$ Unfortunately, I'm not aware of such a reference. In fact I can't figure out any example. On the other hand, you have a full description of the derived category of Abelian varieties, though not through an exceptional collection (if I'm not mistaken). $\endgroup$ – Leo Alonso Aug 31 '18 at 14:39
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    $\begingroup$ Bondal and Polishchuk's "Homological properties of associative algebras: the method of helices." shows that a variety with a full exceptional collection of length $\dim X+1$ must be a fano variety. $\endgroup$ – wnx Aug 31 '18 at 22:41
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Rationality of a variety with a full exceptional collection is a well-know folklore conjecture. In some form a similar open question is mentioned in the paper of Brown and Shipman "The McKay Correspondence, Tilting, and Rationality".

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