If a curve contains a planar subcurve of degree r then its general hyperplane 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 counterexamples. 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 admit 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 contain certain multiple lines. We describe the exceptional curves and give examples of them.
|Number of pages
|Journal of Pure and Applied Algebra
|Published - Nov 8 2001
Bibliographical noteFunding Information:
During the preparation of this paper the authors were also partially supported by the University of Paderborn and the Politecnico di Torino. They thank these institutions for their hospitality.
- Primary 14H45
- Secondary 14H50
ASJC Scopus subject areas
- Algebra and Number Theory