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Sur l’incomplétude de la série linéaire caractéristique d’une famille de courbes planes à nœuds et à cusps

Published online by Cambridge University Press:  22 January 2016

Sébastien Guffroy
Sébastien Guffroy*
Affiliation:
U.F.R. de Mathématiques Bât. M2, Université des Sciences et Technologies de Lille, F59655 Villeneuve d’Ascq Cedex, FRANCE, [email protected]
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Abstract

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Since J.Wahl ([27]), it is known that degree d plane curves having some fixed numbers of nodes and cusps as its only singularities can be represented by a scheme, let say H, which can be singular. In Wahl’s example, H is singular along a subscheme F but the induced reduced scheme Hred is smooth along F. In this work, we construct explicitly a family of plane curves with nodes and cusps which are represented by singular points of Hred.

To this end, we begin to show that the Hilbert scheme of smooth and connected space curves of degree 12 and genus 15 is irreducible and generically smooth. It follows that it is singular along a hypersurface (3.10). This example is minimal in the sense that the Hilbert scheme of smooth and connected space curves is regular in codimension 1 for d < 12 (B.2). Finally we construct our plane curves from the space curves represented by points of this hypersurface (4.7).

Keywords

14B0714H5014N05
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 171 , 2003 , pp. 51 - 83
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2003

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