On the Dimension of the Hilbert Scheme of Curves

Abstract

Consider an irreducible component of the Hilbert scheme whose general point parameterizes a degree d genus g smooth irreducible and nondegenerate curve in a projective variety X. We give lower bounds for the dimension of such components when X is P, P or a smooth quadric threefold in P, respectively. Those bounds make sense from the asymptotic viewpoint if we fix d and let g vary. Some examples are constructed using determinantal varieties to show the sharpness of the bounds for d and g in a certain range. The results can be applied to study rigid curves.