Compactified Jacobians of non-integral curves and Lagrangian fibrations

I report on joint work in progress with Adam Czaplinski, Manfred Lehn, and Sönke Rollenske. We describe moduli spaces of stable sheaves, which are generically line bundles on a certain class of non-integral curves, which we call extended ADE curves. This class generalises the non-integral fibres of elliptic fibrations. The main motivation is that our moduli spaces occur as general singular fibres of Lagrangian fibrations.