Let $X$ be a holomorphic symplectic variety of dimension $2n$ and $\pi: X \to \mathbb{P}^n$ be a Lagrangian fibration. It is known that $\pi$ is smooth outside of the discrimiant divisor $\Delta$. The divisor $\Delta$ is not necessarily irreducible and hence it can be singular. Does somebody know some resctrictions on the singularities of $\Delta$. For example, is $\Delta$ a simple normal crossing divisor?
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$\begingroup$ Do you allow that $(X,[\pi^*\mathcal{O}(1)])$ is sufficiently general in a specified deformation class of Lagrangian fibrations? In that case, there is much more that you can say. $\endgroup$– Jason StarrCommented Dec 18, 2018 at 22:58
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I am afraid not. As it is usual for discriminants, the singularities are rather complicated. Take for instance a K3 surface $S\subset \mathbb{P}^g$, with Picard group generated by $\mathcal{O}_S(1)$. There is a holomorphic symplectic manifold $X$ with a Lagrangian fibration $\pi :X\rightarrow (\mathbb{P}^g)^\vee$, such that the fiber $\pi ^{-1}(h)$, for $h$ a general hyperplane in $\mathbb{P}^g$, is the Jacobian of the curve $h\cap S$. The discriminant locus $\Delta $ is the locus of hyperplanes which are tangent to $S$, that is, the dual hypersurface of $S$. It is irreducible but has quite complicated singularities.
