For Hirzebruch surfaces, the algebraic exceptional set of a three-component normal-crossing curve with big K+B and no fiber or negative self-intersection component is finite and equals the hyper-bitangent set.
Families of curves on a surface of general type.Dokl
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Algebraic Exceptional Set of a Three-Component Curve on Hirzebruch Surfaces
For Hirzebruch surfaces, the algebraic exceptional set of a three-component normal-crossing curve with big K+B and no fiber or negative self-intersection component is finite and equals the hyper-bitangent set.