When does a projective curve contain a planar subcurve?

If a curve contains a planar subcurve of degree r then its general hyperpla ne section contains a subscheme of degree r spanning a line. Here we study to which extent the converse is true. If r is small we describe counterexam ples. However, we give an affirmative answer if r is large with respect to the degree of the curve. This result is achieved by studying curves which a dmit an irreducible two-dimensional family of r-secants. We show that such curves are forced to contain a planar subcurve of degree r unless they cont ain certain multiple lines. We describe the exceptional curves and give exa mples of them. (C) 2001 Elsevier Science B.V. All rights reserved.