2020/06/30 by Bangere, Purnaprajna, Mukherjee, Jayan, Raychaudhury, Debaditya
#14B10 #14D06 #14D15 #14J26 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2006.16448
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 = \mathbbFe parametrized by \mathbb P1, countably many of which are projective. For Y = ℙ2 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 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 \mathbbFe, embedded by a complete linear series are smooth points if and only if 0≤ e≤ 2. In contrast, Hilbert points corresponding to projective K3 carpets supported on ℙ2 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.