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Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it being a member of a base-point-free linear system in a nef-Fano fourfold? What, in anything, is known regarding a similar question in one dimension less (K3 in Fano threefolds)?

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There are some silly, probably useless obstructions, at least in the "honest Fano" case: the Lefschetz hyperplane theorem equate the second Betti and third Betti numbers (resp. Hodge numbers of total weight $2$ and $3$) to the corresponding numbers of the Fano manifold. Fano manifolds of a given dimension are bounded, and explicit examination of the proof should lead to some (probably outrageously large) upper bounds on the Betti numbers. So if the Betti numbers of the CY are larger than those bounds, it cannot be an anticanonical divisor in a Fano.

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    $\begingroup$ There is a mysterious double hyperbola-like curve in the Skarke-Kreuzer paper on hypersurfaces in toric Fano 3folds, which seems to suggest some quadratic inequality on the Hodge numbers, but I wonder if anyone studied it. $\endgroup$ – Lev Borisov Mar 9 '14 at 21:21
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    $\begingroup$ Lev, that curve has fascinated me for a long time. It was even visible in an earlier paper of Candelas et al on hypersurfaces in weighted projective space. To the best of my knowledge, noone has an explanation for it. $\endgroup$ – Mark Gross Mar 10 '14 at 7:06
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For the K3/Fano threefolds case you might have a look at Beauville's paper Fano threefolds and K3 surfaces. Proceedings of the Fano Conference 175-184, Univ. di Torino (2004).

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