TITLE: Necessary condition on Calabi-Yau manfiold to be a hypersurface in a Fano manifold
QUESTION [8 upvotes]: 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)?

REPLY [3 votes]: 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).