On K3 surfaces of Picard rank 14

TL;DR

This paper classifies certain K3 surfaces with Picard rank 14, providing explicit models and describing their moduli spaces, while exploring their relations via the Nikulin construction.

Contribution

It identifies three specific lattices for polarized K3 surfaces of rank 14 and constructs their birational models and moduli spaces, linking them through dual families.

Findings

Explicit birational models as quartic hypersurfaces

Description of moduli spaces using modular invariants

Connection between families via Nikulin construction

Abstract

We study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank-fourteen, 2-elementary lattices. Three such lattices exist, namely $H \oplus E_8(-1) \oplus A_1(-1)^{\oplus 4}$, $H \oplus E_8(-1) \oplus D_4(-1)$, and $H \oplus D_8(-1) \oplus D_4(-1)$. As part of our study, we provide birational models for these surfaces as quartic projective hypersurfaces and describe the associated coarse moduli spaces in terms of suitable modular invariants. Additionally, we explore the connection between these families and dual K3 families related via the Nikulin construction.