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

Changho Keem; Yun-Hwan Kim

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arXiv · 1903.02307

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

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BibTeX RIS

Changho Keem, Yun-Hwan Kim. 2019-03-06. On the Hilbert scheme of linearly normal curves in $\mathbb{P}^4$ of degree $d = g+1$ and genus $g$. https://arxiv.org/abs/1903.02307

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