## Abstract

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.

Original language | English |
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Pages (from-to) | 345-364 |

Number of pages | 20 |

Journal | Journal of Pure and Applied Algebra |

Volume | 164 |

Issue number | 3 |

DOIs | |

State | Published - Nov 8 2001 |

### Bibliographical note

Funding 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.

### Funding

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.

Funders | Funder number |
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Politecnico di Torino | |

Universität Paderborn |

## Keywords

- Primary 14H45
- Secondary 14H50

## ASJC Scopus subject areas

- Algebra and Number Theory