Compactified Jacobians of Extended ADE Curves and Lagrangian Fibrations

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Abstract: We observe that general reducible curves in sufficiently positive linear systems on K3 surfaces are of a form that generalises Kodaira's classification of singular elliptic fibres and thus call them extended ADE curves. On such a curve

C

, we describe a compactified Jacobian and show that its components reflect the intersection graph of

C

. This extends known results when

C

is reduced, but new difficulties arise when

C

is non-reduced. As an application, we get an explicit description of general singular fibres of certain Lagrangian fibrations of Beauville-Mukai type.

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