Automorphism Groups of Certain Enriques Surfaces

Abstract

We calculate the automorphism group of certain Enriques surfaces. The Enriques surfaces that we investigate include very general n-nodal Enriques surfaces and very general cuspidal Enriques surfaces. We also describe the action of the automorphism group on the set of smooth rational curves and on the set of elliptic fibrations.

Mordell-Weil groups and automorphism groups of elliptic K3 surfaces

“…An algorithm on a graph. We recall an algorithm introduced in [5]. Let (V, E) be a simple non-oriented connected graph, where V is the set of vertices and E is the set of edges, which is a set of non-ordered pairs of distinct elements of V :…”

“…Proposition 5.1 (Proposition 4.1 of [5]). The subset V 0 ⊂ V is a complete set of representatives of the orbit decomposition of V by G, and the group G is generated by the union of H and the stabilizer subgroup…”

“…Note that a Conway chamber has infinitely many walls. For the graph (V, E) to have local effectiveness properties in [5], it needs that each L 26 /S X -chamber has only a finite number of walls. We consider the following assumption:…”

“…Thus, under assumptions (5.2) and (5.3), the local effectiveness properties in [5] hold for (V, E) and G, and we can apply the algorithm in [5, Section 4.1] to (V, E) and G.…”

“…In Section 5, we review Borcherds' method. We employ a graph-theoretic formulation of Borcherds' method given in [5,Section 4.1]. Sections 3-5 are intended to be summaries of computational methods in the study of K3 surfaces for future reference.…”