Gonality of a general ACM curve in ℙ³

Robin Hartshorne and Enrico Schlesinger

Vol. 251 (2011), No. 2, 269–313

DOI: 10.2140/pjm.2011.251.269

Abstract

Let C be an ACM (projectively normal) nonsingular curve in ℙ_ℂ³ not contained in a plane, 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 gon(C) − 2.

Keywords

gonality, Clifford index, ACM space curves, multisecant lines

Mathematical Subject Classification 2000

Primary: 14H50, 14H51

Milestones

Received: 11 June 2010

Accepted: 19 October 2010

Published: 3 June 2011