The 2-divisibility of divisors on K3 surfaces in characteristic 2

TL;DR

This paper investigates the divisibility properties of sets of disjoint smooth rational curves on K3 surfaces in characteristic 2, revealing specific configurations possible on supersingular and finite height K3 surfaces.

Contribution

It demonstrates the existence of particular divisibility configurations of rational curves on K3 surfaces in characteristic 2, highlighting differences between supersingular and finite height cases.

Findings

Sets of 8, 12, 16, 20 disjoint rational curves with divisible sum exist on supersingular K3 surfaces.

On K3 surfaces of finite height, only 8 such curves can occur.

Exceptions are at Artin invariants 1 and 10 for supersingular K3 surfaces.

Abstract

We show that K3 surfaces in characteristic 2 can admit sets of $n$ disjoint smooth rational curves whose sum is divisible by 2 in the Picard group, for each $n=8,12,16,20$. More precisely, all values occur on supersingular K3 surfaces, with exceptions only at Artin invariants 1 and 10, while on K3 surfaces of finite height, only $n=8$ is possible.