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On the incompleteness of the characteristic linear series of a family of curves with nodes and cusps. (Sur l’incomplétude de la série linéaire caractéristique d’une famille de courbes planes à nœuds et à cusps.) (French) Zbl 1085.14501
Summary: Since J. M. Wahl [Am. J. Math. 96, 529–577 (1974; Zbl 0299.14008)], it is known that degree \(d\) plane curves having some fixed numbers of nodes and cusps as their only singularities can be represented by a scheme, say \(H\), which can be singular. In Wahl’s example, \(H\) is singular along a subscheme \(F\) but the induced reduced scheme \(H_{\text{red}}\) 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 \(H_{\text{red}}\).
To this end, we begin by showing 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. This example is minimal in the sense that the Hilbert scheme of smooth and connected space curves is regular in codimension 1 for \(d<12\). Finally we construct our plane curves from the space curves represented by points of this hypersurface.

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
14B07 Deformations of singularities
14H50 Plane and space curves
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