K3 surfaces with two involutions and low Picard number

TL;DR

This paper constructs explicit examples of K3 surfaces with two involutions and minimal Picard number over rational numbers, exploring their properties for various degrees and demonstrating the existence of such surfaces with specific Picard lattices.

Contribution

It provides new explicit examples of K3 surfaces with minimal Picard number for degrees 2, 3, and 4, and extends Morrison's result on Picard lattices over the reals.

Findings

Explicit K3 examples with Picard number 2 for degrees 2, 3, 4

Construction of K3 surfaces with prescribed Picard lattices over 

Use of nodal quartic surfaces to realize minimal Picard number for infinitely many degrees

Abstract

Let $X$ be a complex algebraic K3 surface of degree $2d$ and with Picard number $\rho$. Assume that $X$ admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, $\rho \geq 1$ when $d=1$ and $\rho \geq 2$ when $d \geq 2$. For $d=1$, the first example defined over $\mathbb{Q}$ with $\rho=1$ was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kond\={o}, also defined over $\mathbb{Q}$, can be used to realise the minimum $\rho=2$ for all $d\geq 2$. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum $\rho=2$ for $d=2,3,4$. We also show that a nodal quartic surface can be used to realise the minimum $\rho=2$ for infinitely many different values of $d$. Finally, we strengthen a result of Morrison by showing that for any even lattice $N$ of rank $1\leq r \leq 10$ and signature $(1,r-1)$ there exists a K3 surface $Y$ defined over $\mathbb{R}$ such that $\textrm{Pic} Y_\mathbb{C}=\textrm{Pic} Y \cong N$.