Components of the Hilbert Scheme of smooth projective curves using ruled surfaces

Abstract

Let ${\mathcal{I}}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points correspond to smooth irreducible non-degenerate curves of degree $d$ and genus $g$ in ${\mathbb{P}}^r$. We use families of curves on cones to show that under certain numerical assumptions for $d$, $g$ and $r$, the scheme ${\mathcal{I}}_{d,g,r}$ acquires generically smooth components whose general points correspond to curves that are double covers of irrational curves. In particular, in the case $\rho (d,g,r):=g-(r+1)(g-d+r)\ge 0$ we construct explicitly a regular component that is different from the distinguished component of ${\mathcal{I}}_{d,g,r}$ dominating the moduli space ${\mathcal{M}}_g$.