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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

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  • $\begingroup$ I think whenever $D$ is integral, the fiber should be the degree $g$ compactified Picard variety of $D$ which is reduced. I'm not sure what happens above a non-reduced curve but this example is a global version of the Hitchin fibration and I think the analogous local question is what does the Hitchin fiber over a non-reduced spectral curve look like. $\endgroup$ Feb 18, 2023 at 2:43

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