Papers › K3 surfaces with two involutions and low Picard number
K3 surfaces with two involutions and low Picard number
Dino Festi, Wim Nijgh, Daniel Platt
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Let X be a complex algebraic K3 surface of degree 2d and with Picard number ρ. Assume that X admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, ρ≥1 when d=1 and ρ≥2 when d ≥2. For d=1, the first example defined over ℚ with ρ=1 was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kond\={o}, also defined over ℚ, can be used to realise the minimum ρ=2 for all d≥2. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum ρ=2 for d=2,3,4. We also show that a nodal quartic surface can be used to realise the minimum 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≤r ≤10 and signature (1,r-1) there exists a K3 surface Y defined over ℝ such that Pic Y_ℂ=Pic Y ≅N.
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