Algebraic Exceptional Set of a Three-Component Curve on Hirzebruch Surfaces

We study the algebraic exceptional set of a three-component curve $B$ with normal crossings on a Hirzebruch surface $\mathbb{F}_e$. If $K_{\mathbb{F}_{e}}+B$ is big and no component of $B$ is a fiber or the rational curve with negative self-intersection, we prove that the algebraic exceptional set is finite, and in most cases give it an effective bound. We also prove that the algebraic exceptional set coincides with the set of curves that are hyper-bitangent to $B$.