Let $X$ be a Gushel-Mukai threefold, and $S \in |\mathcal{O}_X(H)|$ a hyperplane section of $X$. How are possible hyperplane sections classified?
I know that if $S$ is a generic hyperplane section, then it is a smooth K3 surface of degree $10$. But what are the possibilities when $S$ is singular, i.e., what types of singularities can it have?
In this paper (Lemma 3.2) for del Pezzo threefolds, they show normality by Zak's Theorem on Tangencies. Then, since they take hyperplane sections of a del Pezzo threefold, they quote a classification result for normal Gorenstein surfaces with ample anti-canonical bundle (since the hyperplane section itself is a del Pezzo surface).
Edit #1: But I'm not sure how to apply this line of thinking to the Gushel-Mukai case. By Sasha's answer below (I have now edited this question to account for this), we don't even have normality.
Edit #2: One thing I have in mind is to use this paper. Letting $Y$ for the moment be a Gushel-Mukai surface ($n=2$), we get a $3$-dimensional variety $M_Y = C_K\mathrm{Gr}(2,5) \cap \mathbb{P}(W)$ which by Proposition 2.22 has finitely many rational double points. A hyperplane section of $X$ is exactly a Gushel-Mukai surface. So at least some hyperplane sections of $X$ would look like $Y = M_Y \cap Q(v)$ where $Q(v)$ is a quadric. At least in some (?) cases, cutting by a quadric preserves the singularities.
Does the above reasoning at least find some cases for singular $S$?
