Principal configurations around umbilics of spacelike surfaces in null hypersurfaces of ${\mathbb{R}}_1^4$

We study the principal configurations around an isolated $\eta$-umbilical point on a generic spacelike surface $S$ immersed in a null hypersurface $M$ of Minkowski space ${\mathbb{R}}_1^4$ relative to a well-defined null vector field $\eta$ orthogonal to the surface $S$. In the particular case of $M$ being a null rotation hypersurface of ${\mathbb{R}}_1^4$ we also recover the local Darbouxian principal configurations around that kind of $\eta$-umbilical points.