K3 carpets on minimal rational surfaces and their smoothings

In this paper, we study K3 double structures on minimal rational surfaces [Formula: see text]. The results show there are infinitely many non-split abstract text] parametrized by text], countably of which projective. For exists a unique structure is non-projective (see [J.-M. Drézet, Primitive multiple schemes, preprint (2020), arXiv:2004.04921, to appear in Eur. J. Math.]). We that all projective carpets can be smoothed smooth surface. One the byproducts proof shows unless embedded as variety degree, carpet Moreover, any with arises flat limit embeddings degenerating 2:1 morphism. rest do not, but still prove smoothing result. further Hilbert points corresponding supported complete linear series if and only contrast, (split) always smooth. [P. Bangere, F. Gallego M. González, Deformations hyperelliptic generalized polarized varieties, arXiv:2005.00342] no higher dimensional analogues paper.