On the Hilbert scheme of linearly normal curves in Pr of relatively high degree

Let Hd,g,r be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree d and genus g in Pr. We denote by Hd,g,rL the union of those components of Hd,g,r whose general element is linearly normal and we show that any non-empty Hd,g,rL (d≥g+r−3) is irreducible for an extensive range of triples (d,g,r) beyond the Brill-Noether range. This establishes the validity of a suitably modified assertion of Severi regarding the irreducibility of the Hilbert scheme Hd,g,rL of linearly normal curves for g+r−3≤d≤g+r, r≥3, and g≥2r+3 if d=g+r−3.