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K3 carpets on minimal rational surfaces and their smoothings
International Journal of Mathematics ( IF 0.604 ) Pub Date : 2021-04-07 , DOI: 10.1142/s0129167x21500324
Purnaprajna Bangere, Jayan Mukherjee, Debaditya Raychaudhury

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 1, countably many of which are projective. For Y=2 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 0e2. In contrast, Hilbert points corresponding to projective (split) K3 carpets supported on 2 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:

最小有理曲面上的 K3 地毯及其平滑度

在本文中,我们研究了最小有理曲面上的 K3 双结构 . 结果表明有无数个非分裂的抽象K3双结构在=𝔽电子 参数化 1,其中许多是投射性的。为了=2存在一个独特的非分裂抽象 K3 双结构,它是非投影的(参见 [J.-M. Drézet, Primitive multiple Schemes, preprint (2020), arXiv:2004.04921, to出现在Eur. J. Math. ]) . 我们证明所有投影 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 以各种最小度数嵌入,上面有无限多个嵌入的K3地毯结构 . 此外,我们展示了任何嵌入的投影 K3 地毯𝔽电子电子<3作为退化为 2:1 态射的嵌入的平面限制而出现。其余的没有,但我们仍然证明了平滑结果。我们进一步表明,对应于支持的投影 K3 地毯的希尔伯特点𝔽电子, 嵌入一个完整的线性系列是平滑点当且仅当 0电子2. 相比之下,对应于投影(分裂)K3 地毯的希尔伯特点支持在2和嵌入一个完整的线性系列总是平滑的。结果在 [P. Bangere、FJ Gallego 和 M. González,超椭圆和广义超椭圆极化变体的变形,预印本 (2020),arXiv:2005.00342] 表明本文中的结果没有更高维度的类似物。

更新日期:2021-06-10
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