On lattice-polarized K3 surfaces

TL;DR

This paper refines the definitions of lattice-polarized K3 surfaces, introduces the concept of a 'small cone' for lattice quasipolarization, and describes the moduli space for such surfaces with ADE singularities.

Contribution

It proposes new definitions for lattice-polarized and lattice-quasipolarized K3 surfaces, replacing Weyl chambers with 'small cones', and characterizes their moduli spaces including ADE singularities.

Findings

Introduces the 'small cone' as the key datum in lattice quasipolarization.

Describes the separated moduli stack and space for lattice-polarized K3 surfaces with ADE singularities.

Clarifies the impact of different definitions on the moduli space structure.

Abstract

We propose modifications to the commonly used definitions of lattice-polarized and lattice-quasipolarized smooth K3 surfaces, collecting various versions of the definition, and determining the effects of these choices on the resulting moduli space. We fill a gap in the theory, by replacing Weyl chambers with the new notion of a ``small cone'': the true datum in the definition of lattice quasipolarized K3 surfaces. In addition, we describe the separated moduli stack and moduli space for lattice-polarized K3 surfaces with $ADE$ singularities, an important notion for applications.