On lattice-polarized K3 surfaces

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.