An Algorithm to Compute Automorphism Groups of K3 Surfaces and an Application to Singular K3 Surfaces

Abstract

Abstract. Let X be a complex algebraic K3 surface or a supersingular K3 surface in odd characteristic. We present an algorithm by which, under certain assumptions on X, we can calculate a finite set of generators of the image of the natural homomorphism from the automorphism group of X to the orthogonal group of the Néron-Severi lattice of X. We then apply this algorithm to certain complex K3 surfaces, among which are singular K3 surfaces whose transcendental lattices are of small discriminants. IntroductionThe automorphism group Aut(X) of an algebraic K3 surface X is an important and interesting object. Suppose that X is defined over the complex number field C, or is supersingular in odd characteristic. Then, thanks to the Torelli-type theorem due to and Ogus [21], [22], we can study Aut(X) by the Néron-Severi lattice S X of X. We denote by O(S X ) the orthogonal group of S X . Then we have a natural homomorphism ϕ X : Aut(X) → O(S X ).It is known that this homomorphism has only a finite kernel. Using the reduction theory for arithmetic subgroups of O(S X ), Sterk [36] and Lieblich and Maulik [16] proved that Aut(X) is finitely generated. The Néron-Severi lattices for which Aut(X) are finite were classified by Nikulin [18], [19] and Vinberg [40]. On the other hand, when Aut(X) is infinite, it is in general a difficult problem to give a set of generators.We also have the following related problem. Let Nef(X) denote the nef cone of X; that is, the cone of S X ⊗ R consisting of vectors x ∈ S X ⊗ R such that x, C ≥ 0 holds for any curve C on X, where , is the intersection form on S X . In order to classify various geometric objects on X (for example, smooth rational curves, Jacobian fibrations, or polarizations of a fixed degree) modulo Aut(X), it is useful to describe explicitly a fundamental domain of the action of Aut(X) on Nef(X). Several authors have studied these problems by using the idea of Borcherds [4], [5] to embed S X into an even unimodular hyperbolic lattice of rank 26. In these works, however, they required that S X should satisfy a certain strong condition (see Section 1.1 below for the details), and hence the range of applications is limited.The purpose of this paper is to present an algorithm (Algorithm 6.1) that calculates, under assumptions on S X milder than the preceding works, a finite set of generators of the image of ϕ X and a closed domain F of Nef(X) with the following properties:(i) For any v ∈ Nef(X), there exists an element g ∈ Aut(X) such that v g ∈ F .(ii) The domain F is tiled by a finite number of convex cones, which we call chambers. Each chamber is bounded by a finite number of hyperplanes and its stabilizer subgroup in Aut(X) is finite.See Remark 6.5 for the relation of F with a fundamental domain of the action of Aut(X) on Nef(X). The detailed description of the assumptions we impose on S X will be given in Section 8. The algorithm can be applied to a wide class of K3 surfaces. We give two examples. Example 1.1. As an example of a K3 surface with small Picard number and an i...

The Automorphism Groups of Certain Singular K3 Surfaces and an Enriques Surface

“…In Section 4, we give a computational characterization of the image of the natural homomorphism ϕ X k from Aut(X k ) to O(S k ) and prove Proposition 1.1. In Section 5, we confirm that the requirements to use Borcherds method given in [32] are fulfilled in the cases of our singular K3 surfaces X k , obtain a finite set of generators of Aut(X k ) in the form of matrices in O(S k ) by this method, and prove Theorems 1.4 and 1.5. The embedding of S k into the even unimodular hyperbolic lattice L 26 of rank 26 given in Table 5.1 is the key of this method.…”

“…In fact, a part of the result on Aut(X 2 ) has been obtained in [32]. In [32], however, we did not discuss the problem of converting a matrix in O(S X ) to a geometric automorphism of X. In the present article, we give a method to derive geometric information of automorphisms from their action on S X .…”

Enriques involutions on singular K3 surfaces of small discriminants

“…Finally, the item ε refers to those involutions studied in detail in Section 5.4 and listed in [5]) to calculate aut(X) of a K3 surface X and its action on N X . The details of the algorithms in the computation below are explained in [32]. Suppose that we have a primitive embedding ι X : S X ֒→ L 26 .…”

“…For example, if [ι X ] ⊥ contains a (−2)-vector, then this condition is fulfilled. Condition (B) implies that each ι * X R ⊥ 26 -chamber D in P X has only a finite number of walls (see [32]). More precisely, if D is induced by a Conway chamber C, then the set of vectors defining walls of D can be calculated from the Weyl vector w C corresponding to C by Theorem 2.8.2.…”

“…Let D ∩ (v) be a wall of D, and let D ′ be the ι * X R ⊥ 26 -chamber adjacent to D across D ∩ (v) ⊥ . Then we can calculate the Weyl vector w C ′ corresponding to a Conway chamber C ′ inducing D ′ = ι −1 X (C ′ ) (see [32]), and hence we can calculate the set of walls of D ′ , which is again finite. Therefore we can determine whether there exists an isometry g ∈ O(S X , ω X ) that maps D to D ′ .…”