# A simple formula for the Picard number of K3 surfaces of BHK type

## TL;DR

This paper provides a simplified formula for calculating the Picard number of K3 surfaces of BHK type, showing it depends solely on the degree of the mirror polynomial, advancing understanding of their geometric properties.

## Contribution

The paper simplifies the existing formula for the Picard number of BHK K3 surfaces, revealing it depends only on the degree of the mirror polynomial, which was not previously known.

## Key findings

- Picard number depends only on the degree of the mirror polynomial
- Simplified formula for Picard number of BHK K3 surfaces
- Enhanced understanding of mirror symmetry for K3 surfaces

## Abstract

The BHK mirror symmetry construction stems from work Berglund and Huebsch, and applies to certain types of Calabi-Yau varieties that are birational to finite quotients of Fermat varieties. Their definition involves a matrix $A$ and a certain finite abelian group $G$, and we denote the corresponding Calabi-Yau variety by $Z_{A,G}$. The transpose matrix $A^T$ and the so-called dual group $G^T$ give rise to the BHK mirror variety $Z_{A^T,G^T}$. In the case of dimension 2, the surface $Z_{A,G}$ is a K3 surface of BHK type. Let $Z_{A,G}$ be a K3 surface of BHK type, with BHK mirror $Z_{A^T,G^T}$. Using work of Shioda, Kelly shows that the geometric Picard number of $Z_{A,G}$ may be expressed in terms of a certain subset of the dual group $G^T$. We simplify this formula significantly to show that this Picard number depends only upon the degree of the mirror polynomial $F_{A^T}$.