Extensions of maximal symplectic actions on K3 surfaces

Abstract

We classify pairs (X, G) consisting of a complex K3 surface X and a finite group G Aut(X) such that the subgroup G s G consisting of symplectic automorphisms is among the 11 maximal symplectic ones as classified by Mukai.Résumé. -Nous classifions les paires (X, G) formées d'une surface K3 complexe X et d'un groupe fini G Aut(X) pour lesquelles le sous-groupe G s G des automorphismes symplectiques appartient aux 11 sous-groupes symplectiques maximaux classifiés par Mukai.

K3 surfaces with action of the group M20$M_{20}$

“…We would like to describe its projective model that is in ℙ 17 by Proposition 2.5. By Proposition 2.6, case 3, 𝑋 BH is the intersection of 3 quadrics in ℙ 5 .…”

“…The images of 𝑞 1 , 𝑞 2 and 𝑞 3 via 𝜈 5 2 give three linear equations, thus 𝑞 1 , 𝑞 2 , 𝑞 3 define 3 hyperplanes in ℙ 20 . Their intersection is a 17-dimensional space and in this projective space of dimension 17 the equation of 𝑋 is given by the restrictions of the 105 equations defining 𝜈 5 2 (ℙ 5 ).…”

“…We use the standard notation 𝐸 𝜏 for the elliptic curve ℂ∕(ℤ + 𝜏ℤ) with 𝜏 ∈ ℂ. Proposition 2.6 [see [3,5,13]]. Let 𝐺 be a maximal group acting faithfully on a K3 surface 𝑋 that strictly contain the Mathieu group 𝑀 20 .…”

“…To show the projective model of 𝑋 BH in ℙ 17 we consider the Veronese embedding 𝜈 5 2 ∶ ℙ 5 → ℙ 20 . The image of ℙ 5 via 𝜈 5 2 is the zero locus of quadrics and their number is…”

Finite subgroups of automorphisms of K3 surfaces

“…For instance, one can compute the Néron–Severi and transcendental lattice of a very-general member of the family, the invariant lattice, invariant ample polarizations, the (holomorphic and topological) Euler characteristic of the fixed locus of an automorphism, the isomorphism class of the group, its subgroup consisting of symplectic automorphisms, the number of connected components of the moduli space and the dimension of the moduli space. The pairs with among the maximal groups have been classified in [20]. In many cases, projective models are listed.…”