The gonality of smooth curves with plane models

Abstract

LetC be the normalization of an integral plane curve of degreed with δ ordinary nodes or cusps as its singularities. If δ=0, then Namba proved that there is no linear seriesg ^−2/1 _d and that everyg ^−1/1 _d is cut out by a pencil of lines passing through a point onC. The main purpose of this paper is to generalize his result to the case δ>0. A typical one is as follows: Ifd≥2(k+1), and δ<kd−(k+1)²+3 for somek>0, thenC has no linear seriesg ^−3/1 _d . We also show that ifd≥2k+3 and δ<kd−(k+1)²+2, then each linear seriesg ^−2/1 _d onC is cut out by a pencil of lines. We have similar results forg ^−1/1 _d andg ^−9/1_2d . Furthermore, we also show that all of our theorems are sharp.