Equiclassical deformation of plane algebraic curves

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

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.
Original languageEnglish
Title of host publicationSingularities
Subtitle of host publicationThe Brieskorn Anniversary Volume
EditorsV. I. Arnold, G.-M. Greuel, J. H. M. Steenbrink
PublisherBirkhäuser Basel
Pages195-204
Number of pages10
Volume162
Edition1
ISBN (Electronic)978-3-0348-8770-0
ISBN (Print)978-3-7643-5913-3, 978-3-0348-9767-9
DOIs
StatePublished - 1998

Publication series

NameProgr. Math.
PublisherBirkhäuser, Basel

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