On curves on Hirzebruch surfaces (Q6174638)

In this articles, curves on Hirzebruch surfaces are studied. As a main result, and the inverse to the fact that a Castelnuovo curve lies on a Hirzebruch surface, it is shown that curves of gonality \(k\geq 4\) with sufficiently large genus on Hirzebruch surfaces are Castelnuovo. The other main result of this article shows that a curve on Hirzebruch surface is not a general \(k\)-gonal curve, and in particular the number of the moduli for the curves of fixed gonality \(k\) and of fixed genus \(g\) lying on a Hirzebruch is computed.