Note on a Geometric Isogeny of K3 Surfaces

Abstract

The paper establishes a correspondence relating two specific classes of complex algebraic K3 surfaces. The first class consists of K3 surfaces polarized by the rank-sixteen lattice H ⊕ E7 ⊕ E7. The second class consists of K3 surfaces obtained as minimal resolutions of double covers of the projective plane branched over a configuration of six lines. The correspondence underlies a geometric two-isogeny of K3 surfaces. 1 Geometric Two-Isogenies on K3 Surfaces Let X be an algebraic K3 surface defined over the field of complex numbers. A Nikulin (or symplectic) involution on X is an analytic automorphism of order two Φ: X→ X such that Φ∗(ω) = ω for any holomorphic two-form ω on X. This type of involution has many interesting properties (see [17, 18]), amongst which the most important are: (a) the fixed locus of Φ consists of precisely eight distinct points, and (b) the surface Y obtained as the minimal resolution of the quotient X/Φ is a K3 surface. Equivalently, one can construct Y as follows. Blow up the eight fixed points on X obtaining a new surface X. The Nikulin involution Φ extends to an involution Φ on X which has as fixed locus the disjoint union of the eight resulting exceptional curves. The quotient X/Φ is smooth and recovers the surface Y from above. In the context of the above construction, one has a degree-two rational map pΦ : X 99K Y with a branch locus given by eight disjoint rational curves (the even eight configuration in the sense of Mehran [16]). In addition, there is a push-forward morphism (see [12, 17]) (pΦ)∗ : H(X,Z)→ HY (1) mapping into the orthogonal complement in H(Y,Z) of the even eight curves. The metamorphosis of the surface X into Y is referred to in the literature as the Nikulin construction. The most well-known class of Nikulin involutions is given by the Shioda-Inose structures [12, 17, 18]. These consist of Nikulin involutions that satisfy two additional requirements. The first condition asks for the surface Y to be Kummer. The second requirement asserts that the morphism (1) induces a Hodge isometry between the lattices of transcendental cocycles TX(2) and TY. An effective criterion for a particular K3 surface X to admit a Shioda-Inose structure was given by Morrison [17]. In this paper, we shall work with another class of Nikulin involutions: fiber-wise translations by a section of order two in a jacobian elliptic fibration. This class of involutions was discussed by Van Geemen and Sarti [9]. Let us be precise: Definition 1.1. A Van Geemen-Sarti involution is an automorphism ΦX : X → X for which there exists a triple (φX, S1, S2) such that: (a) φX : X→ P is an elliptic fibration on X, (b) S1 and S2 are disjoint sections of φX, (c) S2 is an element of order two in the Mordell-Weil group MW(φX,S1), (d) ΦX is the involution obtained by extending the fiber-wise translations by S2 in the smooth fibers of φX using the group structure with neutral element given by S1. Under the above conditions, one says that the triple (φX, S1, S2) is compatible with the involution ΦX. Any given Van Geemen-Sarti involution is, in particular, a Nikulin involution. One can naturally regard a Van Geemen-Sarti involution ΦX as a fiber-wise two-isogeny between the original K3 surface X and the newly constructed K3 surface Y. Since ΦX acts as a translation by an element of order two in each of the smooth fibers ∗Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis MO 63108. e-mail: clinghera@umsl.edu †Department of Mathematical and Statistical Sciences, University of Alberta. Edmonton AB T6G 2G1. e-mail: doran@math.ualberta.ca