Classification of Projective Models of $K3$-surfaces

Abstract

Extending the methods previously established for plane sextics and quartic surfaces, we develop a unified and systematic framework for the equisingular deformation classification of projective models of $K3$-surfaces of arbitrary polarization. We treat all polarizations of degree $2n$ with $n\le 4$, including the higher-degree cases of sextic surfaces in ${\mathbb{P}}^4$ and octic surfaces in ${\mathbb{P}}^5$. The computational machinery developed here is degree-independent and provides a complete tool for enumerating and distinguishing the deformation classes arising from singular projective models of $K3$-surfaces. This framework lays the foundation for future investigations in higher degrees, particularly the geometry of equisingular strata, extremal configurations, real structures.