Let $S \to \mathbb P^2$ be a two-to-one cover branched over a sextic, i.e. $S$ is a K3-surface. Let $C \subset S$ be the preimage of a (smooth) quadric, so that by Hurwitz' formula, $g(C) = 5$. According to [1] there is a Lagrangian fibration $$f\colon \mathcal M^s(0, [C], 1) \to |C| \cong \mathbb P^5,$$ where $\mathcal M^s(0, [C], 1)$ denotes the moduli space of stable sheaves on $S$ with Mukai vector $(0, [C], 1)$. If $i\colon D \hookrightarrow S$ is a smooth curve, linearly equivalent to $C$, then the fiber over $D \in |C|$ is $\operatorname{Pic}^g(D)$, given by $L \mapsto i_* L$. So it is smooth
How can I describe the singular fibers of $f$? In particular, how can I describe the singular fibers over the preimage $2E$ of a double line $2L \subset \mathbb P^2$?
My motivation: I found this example in [1], where Sawon writes
the fibres [...] over $2E$ are somewhat like multiple fibres in the theory of elliptic surfaces.
I would like to know if this could produce a (at least local) counterexample to another question. So I wonder if general singular fibers of $f$ are reduced, but those over $2E$ are not?
[1] Justin Sawon, Abelian fibred holomorphic symplectic manifolds, 2003, MR1975339
