Edoardo Ballico, Claudio Fontanari, Changho Keem
Published 2019-04-15Version 1
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 $\mathbb{P}^r$. We let $\mathcal{H}^\mathcal{L}_{d,g,r}$ be the union of those components of $\mathcal{H}_{d,g,r}$ whose general element is linearly normal. In this article, we show that any non-empty $\mathcal{H}^\mathcal{L}_{d,g,r}$ ($d\ge g+r-3$) is irreducible in an extensive range of the triple $(d,g,r)$ beyond the Brill-Noether range. This establishes the validity of a certain modified version of an assertion of Severi regarding the irreducibility of the Hilbert scheme $\mathcal{H}^\mathcal{L}_{d,g,r}$ of linearly normal curves for $g+r-3\le d\le g+r$, $r\ge 5$, and $g \ge 2r+3$ if $d=g+r-3$.
Comments: 10 pages. arXiv admin note: substantial text overlap with arXiv:1903.02307
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