Normal generation of special line bundles on multiple coverings

Abstract

Let X be a smooth curve of genus g which is a multiple covering ϕ : X → Y of a smooth curve Y. We define an invariant Δ ϕ of the map ϕ , which is the maximum among the integers m such that for a base point free line bundle L on X with deg L ≤ g − 1 the inequality Cliff ( L ) ≤ m implies that L is composed with ϕ . Our main result is that a very ample special line bundle L on X with deg L > 3 g − 3 2 is normally generated when Cliff ( L ) ≤ Δ ϕ under additional numerical conditions.