K3 carpets on minimal rational surfaces and their smoothings

Abstract

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