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
$$f: \mathcal M^s(0, [C], 1) \to |C| \cong \mathbb P^5,$$
where $\mathcal M^s(0, [C], 1)$ denotes the moduli space of stable sheaves on $S$ with Mukai vector $(0, [C], 1)$. If $i: D \hookrightarrow S$ 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](https://mathoverflow.net/questions/383422/can-non-reduced-fibers-appear-over-a-subset-of-codimension-geq-2). So I wonder if general singular fibers of $f$ are reduced, but those over $2E$ are not?

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[1] Justin Sawon, *Abelian fibred holomorphic symplectic manifolds*, 2003, [MR1975339](https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=sawon&s5=abelian%20fibred%20holomorphic%20symplectic%20manifolds&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq)