On the Hilbert scheme of smooth curves in ℙ4 of degree d = g + 1 and genus g with negative Brill–Noether number

Abstract

We denote by [Formula: see text] the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree [Formula: see text] and genus [Formula: see text] in [Formula: see text]. In this paper, we show that for low genus [Formula: see text] outside the Brill–Noether range, the Hilbert scheme [Formula: see text] is non-empty whenever [Formula: see text] and irreducible whose only component generically consists of linearly normal curves unless [Formula: see text] or [Formula: see text]. This complements the validity of the original assertion of Severi regarding the irreducibility of [Formula: see text] outside the Brill–Nother range for [Formula: see text] and [Formula: see text].