Compactified Jacobians of Extended ADE Curves and Lagrangian Fibrations

TL;DR

This paper generalizes the classification of singular elliptic fibers to extended ADE curves on K3 surfaces, describing their compactified Jacobians and applying these results to analyze Lagrangian fibrations.

Contribution

It introduces the concept of extended ADE curves and describes their compactified Jacobians, extending known results to non-reduced curves and linking to Lagrangian fibrations.

Findings

Components of the Jacobian reflect the intersection graph of the curve

Extended results for non-reduced curves

Explicit description of singular fibers in Lagrangian fibrations

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.