Overview
This research investigates the configurations of low degree smooth rational curves on quasi-polarized complex K3-surfaces, specifically focusing on K3-sextics with at worst A-D-E singularities. The study applies an efficient approach to classify configurations featuring at least 36 lines. Beyond the classification, the work characterizes a certain class of infinite dihedral groups of birational automorphisms associated with K3-sextics. Additionally, it addresses the impossibility of K3-sextics containing Kummer configurations of lines. A complete account of line configurations on Humbert K3-sextics is also provided, presenting an analysis for this specific analogue.
Research Context
The study contributes to the understanding of configurations of low degree smooth rational curves on (quasi-)polarized complex K3-surfaces. Prior to this work, the specific configurations of lines, especially those involving at least 36 lines on K3-sextics with A-D-E singularities, and certain aspects of their birational automorphisms or line configurations for specific analogues like Humbert K3-sextics, were areas requiring further elucidation.
Approach
The approach employed in this research centers on an efficient methodology for classifying line configurations. This method was applied to identify and categorize configurations involving a minimum of 36 lines on K3-sextics characterized by A-D-E singularities. The study also involved specific characterization processes for certain infinite dihedral groups of birational automorphisms of K3-sextics. Further, a methodical analysis was undertaken to determine the presence or absence of Kummer configurations of lines within K3-sextics. For Humbert K3-sextics, the research involved a comprehensive accounting of their line configurations, treating them as an analogue to Kummer K3-octics or quartics.
Findings
- Configurations of at least 36 lines on K3-sextics with at worst A-D-E singularities were classified.
- A certain class of infinite dihedral groups of birational automorphisms of K3-sextics was characterized.
- No K3-sextic can contain a Kummer configuration of lines.
- A complete account of the line configurations on Humbert K3-sextics was provided.