K3 surfaces over $\mathbb{Q}$ of degree $10$ that have Picard rank $1$

TL;DR

This paper constructs explicit examples of K3 surfaces over the rational numbers with degree 10 and Picard rank 1, expanding the known classifications of such surfaces with minimal Picard groups.

Contribution

It provides explicit geometric constructions of K3 surfaces over  with Picard rank 1, including new examples of degrees 10 and 6.

Findings

Explicit examples of K3 surfaces of degree 10 with Picard rank 1 over .

Construction of a degree 6 K3 surface over  with Picard rank 1.

Geometric descriptions involving intersections in projective space and Grassmannians.

Abstract

We give examples of K3 surfaces over $\mathbb{Q}$ of degree $10$ whose geometric Picard group has rank~$1$. These K3 surfaces are intersections in $\mathbb{P}^9$ of three hyperplanes, one quadric and the image of the Pl\"ucker embedding of the Grasmannian $\mathrm{Gr}(2,5)$. We also give an example of a K3 surface of degree $6$ over~$\mathbb{Q}$ whose Picard rank is $1$.