Hyperplane sections of Calabi-Yau varieties

Theorem. If W is a smooth complex projective variety with h(1) (O-W) = 0, then a sufficiently ample smooth divisor X on W cannot be a hyperplane section of a Calabi-Yau variety, unless W is itself a Calabi-Yau. Corollary. A smooth hypersurface of degree d in P-n (n greater than or equal to 2) is a hyperplane section of a Calabi-Yau variety iff n + 2 less than or equal to d less than or equal to 2n + 2. The method is to construct out of the variety W a universal family of all varieties Z for which X is a hyperplane section with normal bundle K-X, and examine the "bad" singularities of such Z. It was proved in [W1] that if a smooth curve lies on a K-3 surface, its Gaussian-Wahl map Phi(K) is not surjective. Theorem. The following smooth curves do not lie on a K-3, even though Phi(K) is not surjective: plane curves of degree greater than or equal to 7; bielliptic curves of genus greater than or equal to 11; curves on F-n of degree greater than or equal to 5 over P-1.