Quotients of K3 surfaces modulo involutions

Abstract

Abstract. Let X be a K3 surface with an involution σ which has non-empty fixed locus X σ and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, σ) in a canonical way, from some better known double coverings of log del Pezzo surfaces of index ≤ 2 or rational elliptic surfaces, and construct the only family of each of the three extremal case where X σ contains 10 (maximum possible) curves. We also classify rational log Enriques surfaces of index 2.

Classification of K3-surfaces with involution and maximal symplectic symmetry

“…A sharper bound n ≤ 16 for the number of disjoint (−2)-curves on a K3-surface has been obtained by Nikulin [25]. In our setup an even sharper bound is due to Zhang [30,Theorem 3], stating that the total number of connected curves in the fixed point set of an antisymplectic involution on a K3-surface is bounded by 10. In the following, we use Zhang's bound for the number n of rational curves in Fix X (σ ),…”

The alternating groups and K3 surfaces

“…Thus around P, our X h = {y = 0} which is smooth. For the range of s, see [14] or [25]. If X h contains a genus ≥ 2 curve C, then the big and nefness of C and the Hodge index theorem show that the other s − 1 curves are negative definite, whence are P 1 's.…”

On K3 double planes covering Enriques surfaces

“…4], [20]. Apart from the Enriques and the Nikulin type involutions, all other involutions have rational surfaces as quotient.…”

“…The involution μ in addition fixes the nodal classes δ 0 , δ 1 , δ 2 ∈ L given by (20). Let q : X → P = X /ι delP be the natural quotient map.…”