Double covering of curves

Abstract

Let C be a smooth projective algebraic curve of genus q and g an integer with g > 4q + 5. For all integers d > g - 2q + 1 we prove the existence of a double covering f: X → C with X a smooth curve of genus g and the existence of a degree d morphism u: X → P 1 that does not factor through f. By the Castelnuovo-Severi inequality, the result is sharp (except perhaps the bound g > 4q + 5).