Normal generation of line bundles on multiple coverings

A classical result says that a line bundle L on a smooth curve C of genus g with is normally generated. However, it is known that several line bundles of degree d which fail to be normally generated appear on curves of genus g, which are multiple coverings of an algebraic curve, as the degree d is smaller than 2g+1. Thus, investigating the normal generation of line bundles on multiple coverings can be an effective approach to the question of normal generation of line bundles on curves. In this paper, we obtain sufficient conditions for line bundles on a multiple covering to be normally generated and also obtain sufficient conditions for the failure of the property of normal generation.