On two-elementary K3 surfaces with finite automorphism group

TL;DR

This paper classifies certain K3 surfaces with finite automorphism groups, showing they can be modeled as quartic hypersurfaces and detailing their elliptic fibrations and rational curves.

Contribution

It provides a birational model for these K3 surfaces and explicitly constructs all supported Jacobian elliptic fibrations and their geometric features.

Findings

K3 surfaces admit a birational quartic hypersurface model

Explicit construction of all supported Jacobian elliptic fibrations

Determination of dual graphs of rational curves and fiber embeddings

Abstract

We study complex algebraic K3 surfaces of Picard ranks 11,12, and 13 of finite automorphism group that admit a Jacobian elliptic fibration with a section of order two. We prove that the K3 surfaces admit a birational model isomorphic to a projective quartic hypersurface and construct geometrically the frames of all supported Jacobian elliptic fibrations. We determine the dual graphs of all smooth rational curves for these K3 surfaces, the polarizing divisors, and the embedding of the reducible fibers in each frame into the corresponding dual graph.