Rational curves in the moduli of supersingular K3 surfaces

TL;DR

This paper constructs rational curves in the moduli space of supersingular K3 surfaces, revealing that points with Artin invariant 10 lie on infinitely many such curves linked to elliptic structures.

Contribution

It introduces a method to produce non-isotrivial families of supersingular K3 surfaces over rational curves using a relative Artin-Tate isomorphism and twisted Bridgeland moduli spaces.

Findings

Every Artin invariant 10 point lies on infinitely many rational curves.

Constructs explicit non-isotrivial families of supersingular K3 surfaces.

Links rational curves to elliptic structures on K3 surfaces.

Abstract

We show how to construct non-isotrivial families of supersingular K3 surfaces over rational curves using a relative form of the Artin-Tate isomorphism and twisted analogues of Bridgeland's results on moduli spaces of stable sheaves on elliptic surfaces. As a consequence, we show that every point of Artin invariant 10 in the Ogus space of marked supersingular K3 surfaces lies on infinitely many pairwise distinct rational curves canonically associated to elliptic structures on the underlying K3 surface.