Existence and the reducibility of the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ of relatively high degrees

Abstract

Let H d,g,r be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree d and genus g in P r . We denote by H L d,g,r the union of those components of H d,g,r whose general element is linearly normal. In this article we show that H L d,g,r (d ≥ g + r − 3) is non-empty in a wider range of triples (d, g, r) beyond the Brill-Noether range. This settles the existence of the Hilbert scheme H L d,g,r of linearly normal curves in the range g + r − 3 ≤ d ≤ g + r, r ≥ 3 with negative Brill-Noether number, which has been left out almost undetermined in some previous works on the subject. We also determine the full range of triples (d, g, r) with g + r − 3 ≤ d ≤ g + r for which H L d,g,r is reducible or (irreducible).