On the Hilbert scheme of smooth curves of degree $d=15$ in $\mathbb{P}^5$

Edoardo Ballico; Changho Keem

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arXiv · 2310.00682

On the Hilbert scheme of smooth curves of degree $d=15$ in $\mathbb{P}^5$

Edoardo Ballico·Changho Keem

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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 study $\mathcal{H}_{15,g,5}$ for every possible genus $g$ and determine when it is irreducible. We also study the moduli map $\mathcal{H}_{15,g,5}\rightarrow\mathcal{M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements of each component.

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BibTeX RIS

Edoardo Ballico, Changho Keem. 2023-10-01. On the Hilbert scheme of smooth curves of degree $d=15$ in $\mathbb{P}^5$. https://arxiv.org/abs/2310.00682

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