Sur l'incomplétude de la série linéaire caractéristique d'une famille de courbes planes à n?uds et à cusps

摘要

Since J.Wahl ([27]), it is known that degree $d$ plane curves having some fixednumbers of nodes and cusps as its only singularities can be represented by ascheme, let say $H$, which can be singular. In Wahl's example, $H$ is singularalong a subscheme $F$ but the induced reduced scheme $H_{m{red}}$ is smoothalong $F$. In this work, we construct explicitly a family of plane curves withnodes and cusps which are represented by singular points of $H_{m{red}}$.To this end, we begin to show that the Hilbert scheme of smooth and connectedspace curves of degree 12 and genus 15 is irreducible and generically smooth. Itfollows that it is singular along a hypersurface (3.10). This example is minimalin the sense that the Hilbert scheme of smooth and connected space curves isregular in codimension 1 for $d<2$ % (B.2). Finally we construct our planecurves from the space curves represented by points of this hypersurface(4.7).