Reducibility of the Hilbert Scheme of Smooth Curves and Families of Double Covers

Abstract

Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points represent smooth irreducible complex curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. Severi claimed in [15] that $\mathcal{I}_{d,g,r}$ is irreducible if $d \geq g+r$. His statement turned out to be correct for $r = 3$ and $4$, while for $r \geq 6$, counterexamples have been found by using families of $m$-sheeted covers of rational curves with $m \geq 3$. In this work we show the existence of an additional component of $\mathcal{I}_{d,g,r}$ whose general elements are double covers of curves of positive genus. In addition, we find upper bounds for the dimension of the possible components of $\mathcal{I}_{d,g,r}$.