Journal of the
Korean Mathematical Society JKMS

Let $C$ be a smooth $k$-gonal curve of genus $g$. We study the number of pencils of degree $k$ on $C$. In case $g \geq k (k-1)/2$ we state a conjecture based on a discussion on plane models for $C$. From previous work it is known that if $C$ possesses a large number of pencils then $C$ has a special plane model. From this observation the conjectures are split up in two cases: the existence of some types of plane curves should imply the existence of curves $C$ with a given number of pencils; the non-existence of plane curves should imply the non-existence of curves $C$ with some given large number of pencils. The non-existence part only occurs in the range $k(k-1)/2 \leq g \leq k(k-1)/2 + [(k-2)/2]$ if $k \geq 7$. In this range we prove the existence part of the conjecture and we also prove some non-existence statements. Those result imply the conjecture in that range for $k \leq 10$. The cases $k \leq 6$ are handled separately.

Keywords: curves, gonality, pencils, linear systems, plane curves

MSC numbers: 14H45, 14H50, 14H51