## Abstract

We study the geometry of equiclassical families of plane algebraic curves, that means the sets of irreducible curves of a given degree d with given geometric genus g and class c (degree of the dual curve). It is known that the generic member of an equigeneric family (set of curves of given degree with given genus) is a nodal curve, or, in other words, any curve can be deformed into a nodal curve of the same degree and genus. We give sufficient conditions for the existence of a deformation of a plane irreducible curve into a curve of the same degree, genus and class, having only nodes and cusps. For instance, if

c≥2g−d+2

then the generic member of an equiclassical family is a curve with nodes and ordinary cusps, that strengthens the Diaz-Harris sufficient condition c ≥ 2g − 1. In particular, any curve of degree ≤ 10 can be equiclassically deformed into a curve with nodes and cusps.

c≥2g−d+2

then the generic member of an equiclassical family is a curve with nodes and ordinary cusps, that strengthens the Diaz-Harris sufficient condition c ≥ 2g − 1. In particular, any curve of degree ≤ 10 can be equiclassically deformed into a curve with nodes and cusps.

Original language | English |
---|---|

Title of host publication | Singularities |

Subtitle of host publication | The Brieskorn Anniversary Volume |

Editors | V. I. Arnold, G.-M. Greuel, J. H. M. Steenbrink |

Publisher | Birkhäuser Basel |

Pages | 195-204 |

Number of pages | 10 |

Volume | 162 |

Edition | 1 |

ISBN (Electronic) | 978-3-0348-8770-0 |

ISBN (Print) | 978-3-7643-5913-3, 978-3-0348-9767-9 |

DOIs | |

State | Published - 1998 |

### Publication series

Name | Progr. Math. |
---|---|

Publisher | Birkhäuser, Basel |