Clifford Index of Acm Curves in P

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 Clifford index of the curve and the geometrical properties of its embedding. In particular if C is a curve of degree d in P 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 gonality of every smooth ACM curve in P 3 can be computed by multisecants, and we show the answer is yes in some special cases. 1 Gonality and Clifford index of a curve Let C be a nonsingular projective curve over an algebraically closed field k. A linear system of degree d and (projective) dimension r will be denoted by g r d. The least integer d for which there exists a complete linear system g 1 d without base points is called the gonality of C. Thus a curve is rational if and only if its gonality is 1. Curves of genus 1 and 2 have gonality 2. For curves of genus g ≥ 2, the curve is hyperelliptic if and only if the gonality is 2. It is well known that for curves of genus g ≥ 3 the gonality d lies between 2 and g + 3 2 ; there exist curves having each possible gonality in this range; and a curves of genus g of general moduli has gonality g + 3 2. See [1] for references to proofs of these results. Thus the gonality of a curve provides a stratification of the variety of moduli M g of curves of genus g , with the hyperelliptic curves at one end, and the curves of general moduli at the other end.