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arXiv 2006.16448math.AG
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极小有理曲面上的K3地毯及其光滑化

K3 carpets on minimal rational surfaces and their smoothings

Purnaprajna Bangere, Jayan Mukherjee, Debaditya Raychaudhury

备注：
Final version, accepted for publication in International Journal of Mathematics
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摘要：

本文研究极小有理曲面$Y$上的K3重结构。结果表明，在由$\mathbb P^1$参数化的$Y = \mathbb{F}_e$上存在无穷多个非分裂的抽象K3二重结构，其中有可数个是投射的.对于$Y = \mathbb{P}^2$，存在唯一的非分裂抽象K3双结构，它是非投射的（参见Drézet在arXiv：2004.04921中的文章）。我们证明了所有的射影K3地毯都可以被平滑为光滑的K3曲面。证明的一个副产品表明，除非$Y$被嵌入为各种最小程度，否则在$Y$上有无限多个嵌入的K3地毯结构。此外，我们发现任何嵌入的投影K3地毯$\mathbb F_e$与$e<3$作为一个平坦的极限退化到$2:1$态射的嵌入。其余的不做，但我们仍然证明了平滑的结果。进一步证明了由完备线性级数嵌入的投影K3地毯所对应的Hilbert点是光滑点的充要条件是$\mathbb{F}_e$.相反，对应于支持在$\mathbb{P}^2$上并被完全线性级数嵌入的投射K3地毯的Hilbert点总是光滑的。Bangere、Gallego和González最近的一篇论文中的结果表明，本文中的结果没有高维类似物。

英文摘要：

In this article, we study K3 double structures on minimal rational surfaces $Y$. The results show there are infinitely many non-split abstract K3 double structures on $Y = \mathbb{F}_e$ parametrized by $\mathbb P^1$, countably many of which are projective. For $Y = \mathbb{P}^2$ there exist a unique non-split abstract K3 double structure which is non-projective (see Drézet's article in arXiv:2004.04921). 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 $\mathbb F_e$ 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 $\mathbb{F}_e$, embedded by a complete linear series are smooth points if and only if $0\leq e\leq 2$. In contrast, Hilbert points corresponding to projective K3 carpets supported on $\mathbb{P}^2$ and embedded by a complete linear series are always smooth. The results in a recent paper of Bangere, Gallego, and González show that there are no higher dimensional analogues of the results in this article.

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