On the Hilbert scheme of curves of maximal genus in a projective space
Given two integers d, r, with \(d\geq r\geq 3\), there is an upper bound, discovered by Castelnuovo in 1883, for the arithmetic genus of an irreducible, nondegenerate curve of degree d in \(P^ r\). Curves for which the bound is attained are called Castelnuovo curves. This paper deals with the study of some local properties of the Hilbert scheme at points corresponding to smooth Castelnuovo curves. The main result concerns a geometrical characterization of obstructed Castelnuovo curves, namely smooth Castelnuovo curves for which the corresponding point in the Hilbert scheme is singular. These turn out to be curves in \(P^ r\), \(r\geq 4\), lying on rational normal cones, and, if \(r=4\), they are in fact complete intersection curves on a rational normal cubic cone. The proof consists in explicitly computing the dimension of each component of the Hilbert scheme of such curves, and in comparing it with the dimension of the Zariski tangent space. The first may be done by means of geometrical arguments. In order to know the latter one has, of course, to compute the cohomology of the normal bundle to smooth Castelnuovo curves, and this is worked out with techniques which may be of some use in other similar questions.
Hilbert scheme at points corresponding to smooth Castelnuovo curves
scientific article; zbMATH DE number 3957275