Pencils on double coverings of curves

Abstract

Let X be a smooth curve of genus g. When \(\pi \geq 3g\) and d ≥ π−2g+1 we show the existence of a double covering \(\gamma : C \rightarrow X\) where C a smooth curve of genus π with a base-point-free pencil of degree d which is not the pull-back of a pencil on X.