D'autres composantes non réduites de Hilb\({\mathbb{P}}^ 3\). (Other non reduced components of Hilb\({\mathbb{P}}^ 3\))
The first example of a non-reduced component of the Hilbert scheme corresponding to nonsingular space curves was constructed by \textit{D. Mumford} [Am. J. Math. 84, 642-648 (1962; Zbl 0114.131)]. The topic was later studied by \textit{J. Kleppe} (Thesis, Oslo 1982), who constructed a series of such components, the general curves of which are contained in a cubic surface. The present paper aimes at a conjecture by Kleppe on necessary and sufficient conditions for a family of curves on cubic surfaces to be such a non-reduced component of the Hilbert scheme. While pointing out that Kleppe's conjecture must be modified a little (adding the hypothesis of linear normality in certain cases), the author is able to prove the conjecture in a wider range of degree and genus than was known before, although not the full range. To show that a given curve lies on a non-reduced component, one finds first the dimension of the tangent space to the Hilbert scheme by deformation theory. If the curve moves in a family of strictly smaller dimension, we are basically done. The hard part, therefore, is to verify that a given curve on a cubic is not a specialization of a curve not on a cubic. In the cases treated in this paper, the relative size of degree and genus guarantees that the general curve is at least on a quartic, and the proof is by a study of curves on quartic surfaces.
non-reduced component of the Hilbert scheme
scientific article; zbMATH DE number 4033887
scientific article