2017/02/13 by Lev Borisov, Borisov, Lev
Mathematics · #14J28 #14J33 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1702.03919
openalex publication_date 2017/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a two-parameter family of K3 surfaces of (generic) Picard rank 18 which is mirror to the 18-dimensional family of elliptically fibered K3 surfaces with a section. Members of this family are given as compactifications of hypersurfaces in three dimensional algebraic torus given by equations y2+z+z-1 + x3 + ax +b =0 where a and b are the parameters of the family. It follows from previous work of Morrison that these surfaces are double covers of Kummer surfaces coming from products of elliptic curves, and we establish this connection explicitly. This in turn provides a description of the one-parameter families of K3 surfaces which are mirror to polarized K3 surfaces of Picard rank one. It has been brought to our attention that the results of this paper have already appeared in the literature. The appropriate references are added at the end of the introduction.