The Severi line equality, already known in characteristic zero, is now characterized in all characteristics: equality holds exactly for canonical models that are flat double covers of Abelian surfaces.
and Stoppino, L.: Surfaces on the Severi line , Journal de Math´ ematiques Pures et Appliqu´ ees,105 no
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Surfaces on the Severi line in positive characteristics
The Severi line equality, already known in characteristic zero, is now characterized in all characteristics: equality holds exactly for canonical models that are flat double covers of Abelian surfaces.