Let $X$ be a Picard rank $1$, index $1$, degree $10$ Fano threefold with polarization $H$. Let $S \subset X$ be a hypersurface with polarization $H_S$ (so $S$ is a degree $10$ K3 surface). Let $D \subset S$ be a divisor, such that $H_S \cdot D = -1$, with the following possibilities for its self-intersection number: either $D^2 = -4$, $-2$, or $0$.
My question is: what can we say explicitly about $D$ in each of the three cases, and which of them are even possible (considering that $S$ is a degree $10$ K3 surface, etc.)?