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

TL;DR

This paper investigates the algebraic exceptional set of a three-component curve with normal crossings on Hirzebruch surfaces, establishing finiteness and bounds under certain conditions, and relating it to hyper-bitangent curves.

Contribution

It proves the finiteness of the algebraic exceptional set and provides effective bounds, also showing its equivalence to the set of hyper-bitangent curves in specific cases.

Findings

The algebraic exceptional set is finite under given conditions.

Effective bounds for the algebraic exceptional set are established.

The exceptional set coincides with hyper-bitangent curves in most cases.

Abstract

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 $B$.