On the Hilbert scheme of linearly normal curves in ${\mathbb{P}}^4$ of degree $d=g+1$ and genus $g$

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 ${\mathbb{P}}^r$. In this article, we show that any non-empty ${\mathcal{H}}_{g+1,g,4}$ has only one component whose general element is linear normal unless $g=9$. If $g=9$, we show that ${\mathcal{H}}_{g+1,g,4}$ is reducible with two components and a general element of each component is linearly normal. This establishes the validity of a certain modified version of an assertion of Severi regarding the irreducibility of ${\mathcal{H}}_{d,g,r}$ for the case $d=g+1$ and $r=4$.

Correction of a few misprints. Final version; to appear in Arch. Math