Existence and reducibility of the Hilbert scheme of smooth and linearly normal curves in Pr 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 d , g , r L the union of those components of H d , g , r whose general element is linearly normal. In this article we show that H d , g , r L ( d ≥ g + r − 3 ) is non-empty in a certain optimal range of triples ( d , g , r ) and is empty outside the range. This settles the existence (or non-emptiness if one prefers) of the Hilbert scheme H d , g , r L of linearly normal curves of degree d and genus g in P r for g + r − 3 ≤ d ≤ g + r , r ≥ 3 . We also determine all the triples ( d , g , r ) with g + r − 3 ≤ d ≤ g + r for which H d , g , r L is reducible (or irreducible).