Rational curves on the supersingular K3 surface with Artin invariant 1 in characteristic 3

TL;DR

This paper demonstrates the existence of numerous rational curves on a specific supersingular K3 surface in characteristic 3, revealing intricate geometric configurations and connections to lattice theory.

Contribution

It establishes the existence of 112 rational curves on the surface and explores their relation to Leech lattice roots, providing new geometric and lattice-theoretic insights.

Findings

112 non-singular rational curves exist on the surface

Configurations of rational curves are identified and described

The Picard lattice is analyzed via Leech lattice theory

Abstract

We show the existence of 112 non-singular rational curves on the supersingular K3 surface with Artin invariant 1 in characteristic 3 by several ways. Using these rational curves, we have a $(16)_{10}$-configuration and a $(280_{4}, 112_{10})$-configuration on the K3 surface. Moreover we study the Picard lattice by using the theory of the Leech lattice. The 112 non-singular rational curves correspond to 112 Leech roots.