The gonality of curves on a Hirzebruch surface
Let \(X_e\) be a Hirzebruch surface, that is a geometrically ruled rational surface of invariant \(e\geq 0\). The main theorem in this paper shows that for \(C\) a reduced and irreducible curve on \(X_e\), which is not a fibre, the minimal degree \(k\) of a morphism from \(C\) to \(\mathbb{P}^1\) is computed (with one exception) by a ruling of \(X_e\). In the case where \(C\) is smooth \(k\) is the gonality of \(C\), that is the smallest degree of a linear pencil on \(C\). The author applies his main result to calculate the gonality of smooth irreducible curves \(C\) on \(X_e\). The author uses an inequality of Serrano [cf. \textit{F. Serrano}, Math. Ann. 277, 395-413 (1987; Zbl 0595.14005)] and \textit{R. Paoletti}, Math. Ann. 303, No. 1, 109-123 (1995; Zbl 0835.14005)] to prove his main result.
scientific article; zbMATH DE number 938612