On the Dimension of the Hilbert Scheme of Curves

Consider an irreducible component of the Hilbert scheme whose general points parameterize degree $d$ genus $g$ smooth irreducible and non-degenerate curves in a projective variety $X$. We give lower bounds for the dimension of such components when $X$ is $\mathbb P^3, \ \mathbb P^4$ or a smooth quadric threefold in $\mathbb P^4$, 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 be applied to study rigid curves.

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Published 1 January 2009