Gonality of a general ACM curve in projective 3-space

TL;DR

This paper determines the gonality of a general ACM curve in projective 3-space, linking it to the degree and multisecant lines, and explores related invariants like the Clifford index.

Contribution

It proves the gonality equals the degree minus the maximum multisecant line order for general ACM curves, addressing a question by Peskine and analyzing special cases.

Findings

Gonality equals d - l, with l being the maximum multisecant line order.

l=4 in most cases, with exceptions characterized by the curve's postulation.

Clifford index equals gonality minus 2.

Abstract

Let C be an ACM (projectively normal) nondegenerate smooth curve in projective 3-space, and suppose C is general in its Hilbert scheme - this is irreducible once the postulation is fixed. Answering a question posed by Peskine, we show the gonality of C is d-l, where d is the degree of the curve, and l is the maximum order of a multisecant line of C. Furthermore l=4 except for two series of cases, in which the postulation of C forces every surface of minimum degree containing C to contain a line as well. We compute the value of l in terms of the postulation of C in these exceptional cases. We also show the Clifford index of C is equal to the gonality minus 2.