K3 surfaces with two involutions and low Picard number

Abstract

AbstractLet X be a complex algebraic K3 surface of degree 2d and with Picard number $$\rho $$ ρ . Assume that X admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, $$\rho \ge 1$$ ρ ≥ 1 when $$d=1$$ d = 1 and $$\rho \ge 2$$ ρ ≥ 2 when $$d \ge 2$$ d ≥ 2 . For $$d=1$$ d = 1 , the first example defined over $${\mathbb {Q}}$$ Q with $$\rho =1$$ ρ = 1 was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over $${\mathbb {Q}}$$ Q , can be used to realise the minimum $$\rho =2$$ ρ = 2 for all $$d\ge 2$$ d ≥ 2 . In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum $$\rho =2$$ ρ = 2 for $$d=2,3,4$$ d = 2 , 3 , 4 . We also show that a nodal quartic surface can be used to realise the minimum $$\rho =2$$ ρ = 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 $$1\le r \le 10$$ 1 ≤ r ≤ 10 and signature $$(1,r-1)$$ ( 1 , r - 1 ) there exists a K3 surface Y defined over $${\mathbb {R}}$$ R such that $${{\,\textrm{Pic}\,}}Y_{\mathbb {C}}={{\,\textrm{Pic}\,}}Y \cong N$$ Pic Y C = Pic Y ≅ N .