  Consider a component of the Hilbert scheme whose general point corresponds to
a degree d genus g smooth irreducible and nondegenerate curve in a projective
variety X. We give lower bounds for the dimension of such a component when X is
P^3, P^4 or a smooth quadric threefold in P^4 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 also be applied to study
rigid curves.
