Normal generation of line bundles on multiple coverings

Abstract

A classical result says that a line bundle L on a smooth curve C of genus g with deg L 2g + 1 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.

Explicit presentations of nonspecial line bundles and secant spaces

“…Corollary 2.5 (Lemma 6, [9]). Let L be a nonspecial line bundle on a smooth curve X which is presented by K X − g 0 d + E with degE ≥ 3.…”

Gonality and Clifford index of projective curves on ruled surfaces

“…In fact, if the covering morphism is simple, then we can apply the Castelnuovo-Severi inequality: it bounds the degree of line bundles which are not composed with the the covering morphism. Thus, in the cases where the covering morphisms are simple, there are some theorems to determine whether a given line bundle is or is not composed with φ ( [2], [11], [15], [10]).…”