Idoneal genera and K3 surfaces covering an Enriques surface

Abstract

We classify the transcendental lattices of all K3 surfaces covering an Enriques surface. In order to do so, we classify all genera of positive definite lattices that only contain lattices with a vector of square 1.

Symplectic birational transformations of finite order on O'Grady's sixfolds

“…If Λ is a unimodular lattice and G ⊂ O(Λ) is a cyclic subgroup of order m, then Λ G and Λ G are m-elementary lattices.The following statements are a direct consequence of (5) and[5, Lemma 3.1].Lemma 2.6. If M is an m-elementary lattice, L is an l-elementary lattice and there exists a primitive embedding M ֒→ L, then N is lcm(m, l)-elementary.Lemma 2.7.…”

“…Known deformation types τ of irreducible holomorphic symplectic manifolds (n > 1). 3 6 8 3U ⊕ 2[−2] OG 5 10 24 3U ⊕ 2E 8 ⊕ A 2…”

Enriques involutions on pencils of K3 surfaces

“…Table 1. Lattices in the frame genus W X of a K3 surface X with transcendental lattice T X ∼ = U ⊕ [4].…”

“…Its transcendental lattice T X is an even lattice of signature (2,1). By a result of Brandhorst, Sonel and the second author [4], the surface X covers an Enriques surface only if 4 divides det(T X ), but this condition is not sufficient. In this paper we analyze in detail what happens when |det(T X )| is small, more precisely (1) |det(T X )| < 16.…”

“…Families I and II are 1-dimensional, while families III to VII are 0-dimensional. The K3 surfaces covering the generic Enriques surface of type I and II have transcendental lattice T X ∼ = U ⊕ [4] and T X ∼ = U ⊕ [8], respectively.…”