Irreducibility of the Hilbert scheme of smooth curves in $\Bbb P^4$ of degree $g+2$ and genus $g$

Changho Keem; Yun-Hwan Kim

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arXiv · 1703.07516

Irreducibility of the Hilbert scheme of smooth curves in $\Bbb P^4$ of degree $g+2$ and genus $g$

Changho Keem·Yun-Hwan Kim

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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 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.

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BibTeX RIS

Changho Keem, Yun-Hwan Kim. 2017-03-22. Irreducibility of the Hilbert scheme of smooth curves in $\Bbb P^4$ of degree $g+2$ and genus $g$. https://arxiv.org/abs/1703.07516

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