We study the Hilbert scheme $\mathcal{ H}^\mathcal{ L}_{ d,g,r}$ parametrizing smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in $\mathbb{ P}^r$ whose complete and very ample hyperplane linear series $\mathcal{ D}$ have relatively small index of speciality $i(\mathcal{ D})=g-d+r$. In particular we show the existence (and non-existence as well in some sporadic cases) of every Hilbert scheme of linearly normal curves with $i(\mathcal{ D})=4$. We also determine the irreducibility of $\mathcal{ H}^\mathcal{ L}_{ 2r+4,r+8,r}$ for $3\le r\le 8$, which are rather peculiar families in a certain sense.
Categories: math.AG