TY - JOUR

T1 - Compactified Jacobians of Extended ADE Curves and Lagrangian Fibrations

AU - Czapliński, Adam

AU - Krug, Andreas

AU - Lehn, Manfred

AU - Rollenske, Sönke

PY - 2024/3/9

Y1 - 2024/3/9

N2 - 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.

AB - 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.

KW - math.AG

U2 - 10.48550/arXiv.2206.11686

DO - 10.48550/arXiv.2206.11686

M3 - Article

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

ER -