Clifford Index of ACM Curves in ${\mathbb P}^3$

Abstract

In this paper we review the notions of gonality and Clifford index of an abstract curve. For a curve embedded in a projective space, we investigate the connection between the \ci of the curve and the \gc al properties of its \emb. In particular if $C$ is a curve of degree $d$ in $¶^3$, and if $L$ is a multisecant of maximum order $k$, then the pencil of planes through $L$ cuts out a $g^1_{d-k}$ on $C$. If the gonality of $C$ is equal to $d-k$ we say the gonality of $C$ can be computed by multisecants. We discuss the question whether the \go of every smooth ACM curve in $¶^3$ can be computed by multisecants, and we show the answer is yes in some special cases.