As an application of a nice theorem of Serrano ([4], [3]) we compute the gonality gon(C) (= the smallest degree of a linear pencil) of a smooth irreducible curve C defined over C and lying on a Hirzebruch surface Xe, i.e. a geometrically ruled rational surface of invariant e 0 ([1], V, 2.13). Recall, Pic (Xe) is generated by the classes of two rational curves Co, F where Co is a zero-section (Cg = -e) and F is a fibre of the bundle map ft- Xe -> P1. For e > 0 the fibres of n constitute the only ruling (= pencil of smooth irreducible rational curves with self-intersection number zero) of Xe whereas XQ has exactly two rulings (the pencils |F| and |C0|, of course).
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