Let C be an ACM
(projectively normal) nonsingular curve in ℙℂ3 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