On the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with small index of speciality

Abstract

We study the Hilbert scheme of smooth, irreducible, nondegenerate and linearly normal curves of degree d and genus g in P r 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 the non-existence of certain Hilbert schemes with i(D) = 4. We also determine the irreducibility of H L 2r+4,r+8,r for 3 ≤ r ≤ 8, which are rather peculiar families in some sense.