On the Hilbert scheme of linearly normal curves in [formula omitted] with small index of speciality

Abstract

We study the Hilbert scheme Hd,g,rL parametrizing smooth, irreducible, non-degenerate and linearly normal curves of degree d and genus g in Pr whose complete and very ample hyperplane linear series D have relatively small index of speciality i(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(D)=4. We also determine the irreducibility of H2r+4,r+8,rL for 3≤r≤8, which are rather peculiar families in a certain sense.