On some components of Hilbert schemes of curves

Abstract

Let $\mathcal{I}_{d,g,R}$ be the union of irreducible components of the Hilbert scheme whose general points parametrize smooth, irreducible, curves of degree $d$, genus $g$, which are non--degenerate in the projective space $\mathbb{P}^R$. Under some numerical assumptions on $d$, $g$ and $R$, we construct irreducible components of $\mathcal{I}_{d,g,R}$ other than the so-called {\em distinguished component}, dominating the moduli space $\mathcal{M}_g$ of smooth genus--$g$ curves, which are generically smooth and turn out to be of dimension higher than the expected one. The general point of any such a component corresponds to a curve $X \subset \mathbb{P}^R$ which is a suitable ramified $m$--cover of an irrational curve $Y \subset \mathbb{P}^{R-1}$, $m \geqslant 2$, lying in a surface cone over $Y$. The paper extends some of the results in previous papers of Y. Choi, H. Iliev, S. Kim (cf. [12,13] in Bibliography).