On the Hilbert scheme of linearly normal curves in P-r with small index of speciality

Abstract

We study the Hilbert scheme H-d,g,r(L) parametrizing smooth, irreducible, non-degenerate 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 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 H-2r+4,H-r+8,(L)(r) for 3 <= r <= 8, which are rather peculiar families in a certain sense. (C) 2022 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.