K3 carpets on minimal rational surfaces and their smoothings

TL;DR

This paper investigates K3 double structures on minimal rational surfaces, demonstrating their existence, smoothing properties, and the conditions under which their Hilbert points are smooth, with implications for algebraic geometry and surface theory.

Contribution

It characterizes the existence and smoothing of K3 carpets on minimal rational surfaces, including explicit parametrizations and conditions for smoothness of Hilbert points.

Findings

Infinitely many non-split K3 double structures on $\

$ ext{Y} = ext{F}_e$ are parametrized by $\

Abstract

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 Dr\'ezet'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 Gonz\'alez show that there are no higher dimensional analogues of the results in this article.