Extendability of general $K3$ surfaces without Gaussian maps and classification of non-prime Fano threefolds

In arXiv:2409.03960, we introduced an approach to the question of extendability of projective varieties via degeneration to ribbons. In this article we build on these methods to give a new proof of optimal results on the extendability of general non-prime $K3$ surfaces, classification of non-prime Fano threefolds and Mukai varieties and the irreducibility of their Hilbert schemes. The methods in this article also show the non-extendability of prime $K3$ surfaces for infinitely many values of $g$, for example when $g$ is of the form $g=4k+1$, $k\ge 5$. This involves degenerations of $K3$ surfaces to ribbons on embedded Hirzebruch surfaces, called $K3$ carpets. We directly give optimal upper bounds on the cohomology of the twisted normal bundle of the $K3$ carpets instead of computing coranks of Gaussian maps of the canonical curve sections. As a result of independent interest, we show such $K3$ carpets also appear as degenerations of smoothable simple normal crossings of two Hirzebruch surfaces embedded by arbitrary linear series intersecting along an anticanonical elliptic curve. Such type II degenerations constitute a smooth locus of codimension $6$ in the Hilbert scheme of $K3$ surfaces.