Changho Keem, Yun-Hwan Kim
Published 2017-03-22Version 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 $\PP^r$. In this note, we show that any non-empty $\mathcal{H}_{g+2,g,4}$ is irreducible without any restriction on the genus $g$. Our result augments the irreducibility result obtained earlier by Hristo Iliev(2006), in which several low genus $g\le 10$ cases have been left untreated.
Comments: Any critical comments are welcome
On the Hilbert scheme of smooth curves of degree $d=15$ in $\mathbb{P}^5$
On the Hilbert scheme of smooth curves of degree $15$ and genus $14$ in $\mathbb{P}^5$
On the Hilbert scheme of smooth curves in $\mathbb{P}^4$ of degree $d = g+1$ and genus $g$ with negative Brill-Noether number
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