Extendability of projective varieties via degeneration to ribbons with applications to Calabi-Yau threefolds

In this article we study the extendability of a smooth projective variety by degenerating it to a ribbon. We apply the techniques to study extendability of Calabi-Yau threefolds $X_t$ that are general deformations of Calabi-Yau double covers of Fano threefolds of Picard rank $1$. The Calabi-Yau threefolds $X_t \hookrightarrow \mathbb{P}^{N_l}$, embedded by the complete linear series $|lA_t|$, where $A_t$ is the generator of Pic$(X_t)$, $l \geq j$ and $j$ is the index of $Y$, are general elements of a unique irreducible component $\mathscr{H}_l^Y$ of the Hilbert scheme which contains embedded Calabi-Yau ribbons on $Y$ as a special locus. For $l = j$, using the classification of Mukai varieties, we show that the general Calabi-Yau threefold parameterized by $\mathscr{H}_j^Y$ is as many times smoothly extendable as $Y$ itself. On the other hand, we find for each deformation type $Y$, an effective integer $l_Y$ such that for $l \geq l_Y$, the general Calabi-Yau threefold parameterized by $\mathscr{H}_l^Y$ is not extendable. These results provide a contrast and a parallel with the lower dimensional analogues; namely, $K3$ surfaces and canonical curves, which stems from the following result we prove: for $l \geq l_Y$, the general hyperplane sections of elements of $\mathscr{H}_l^Y$ fill out an entire irreducible component $\mathscr{S}_l^Y$ of the Hilbert scheme of canonical surfaces which are precisely $1-$ extendable with $\mathscr{H}^Y_l$ being the unique component dominating $\mathscr{S}_l^Y$. The contrast lies in the fact that for polarized $K3$ surfaces of large degree, the canonical curve sections do not fill out an entire component while the parallel is in the fact that the canonical curve sections are exactly one-extendable.