In this paper, we study K3 double structures on minimal rational surfaces Y. The results show there are infinitely many non-split abstract K3 double structures on Y = 𝔽e parametrized by ℙ1, countably many of which are projective. For Y = ℙ2 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 Y is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on Y. Moreover, we show any embedded projective K3 carpet on 𝔽e with e < 3 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 𝔽e, embedded by a complete linear series are smooth points if and only if 0 ≤ e ≤ 2. In contrast, Hilbert points corresponding to projective (split) K3 carpets supported on ℙ2 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.

中文翻译：

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最小合理表面上的 K3 地毯及其平滑度

在本文中，我们研究了最小有理面上的 K3 双结构是. 结果表明，在是 = 𝔽e参数化ℙ1, 可数其中许多是射影的。为了是 = ℙ2存在一个独特的非分裂抽象 K3 双重结构，它是非投射的（参见 [J.-M. Drézet, Primitive multiple scheme, preprint (2020), arXiv:2004.04921, to appear in欧元。J.数学。]）。我们展示了所有投影 K3 地毯都可以平滑到光滑的 K3 表面。证明的副产品之一表明，除非是作为各种最小程度的嵌入，有无数个嵌入的K3地毯结构是. 此外，我们展示了任何嵌入式投影 K3 地毯𝔽e和e < 3出现作为嵌入的平坦限制退化为 2:1 态射。其余的没有，但我们仍然证明了平滑结果。我们进一步表明，希尔伯特点对应于支持的投影 K3 地毯𝔽e, 嵌入一个完整的线性级数是光滑点当且仅当0 ≤ e ≤ 2. 相比之下，希尔伯特点对应于支持的投影（分裂）K3 地毯ℙ2并且嵌入一个完整的线性级数总是平滑的。[P. Bangere、FJ Gallego 和 M. González，超椭圆和广义超椭圆偏振变种的变形，预印本 (2020)，arXiv:2005.00342] 表明本文中的结果没有更高维的类似物。