Real Structures in the Moduli of Projective Models of K3-Surfaces

Abstract

This paper investigates when a complex family of K3-surfaces with prescribed simple singularities contains real algebraic surfaces. We develop an explicit and uniform algorithm for detecting real representatives in any equisingular stratum, valid for all polarizations. As a primary application, we resolve the problem for spatial quartics, showing that all real equisingular strata contain real surfaces except for three exceptional types, thereby completing the study of real representability in this case. Our method also revisits the analogous phenomenon in the realm of plane sextics, recovering the unique known real stratum of simple sextics without real representatives.