Lagrangian Fibrations with designed singular fibres

In this talk, we study Lagrangian Fibrations with designed singular fibers. The idea is to construct a $K3$ surface $X$ as a minimal resolution of the singularities of a double cover $Y$ of the plane branched along a reduced but possibly reducible singular sextic $\Sigma$. Moreover, we assume that $\Sigma$ has at worst $A$-$D$-$E$ singularities. This freeness of choosing $\Sigma$ allows us to construct many examples of singular fibres with various singularities.

We find an explicit description of the singular fibers of the Lagrangian Fibrations $f\colon M_X(0,2H,\chi)\rightarrow |2H|$. The results shed also some light on the correlation between the degree of the discriminant divisor $\Delta$ and the topology of the corresponding moduli space.