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 ${\mathcal {M}}_g$. Our result implies also that if $g \ge 57$ then ${\mathcal {I}}_{\frac{4g}{3}, g, \frac{g+1}{2}}$ has at least two generically smooth components parametrizing linearly normal curves.