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Abstract

We denote by $\mathcal{H}_{d,g,r}$ 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 $d$ and genus $g$ in $\PP^r$. In
this article, we show that for low genus $g$ outside the Brill-Noether range,
the Hilbert scheme $\mathcal{H}_{g+1,g,4}$ is non-empty whenever $g\ge 9$ and
irreducible whose only component generically consists of linearly normal curves
unless $g=9$ or $g=12$. This complements the validity of the original assertion
of Severi regarding the irreducibility of $\mathcal{H}_{d,g,r}$ outside the
Brill-Nother range for $d=g+1$ and $r=4$.