References & Citations
Mathematics > Algebraic Geometry
Title: Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with index of speciality five and beyond
(Submitted on 3 Dec 2023 (v1), last revised 7 Feb 2024 (this version, v2))
Abstract: We study the Hilbert scheme of smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ ($r\ge 3$) whose complete and very ample hyperplane linear series $\mathcal{D}$ have relatively small index of speciality $i(\mathcal{D})=g-d+r$. In particular we determine the existence as well as the non-existence of Hilbert schemes of linearly normal curves $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ for every possible triples $(d,g,r)$ with $i(\mathcal{D})=5$ and $r\ge 3$. We also determine the irreducibility of the Hilbert scheme $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$ when the genus $g$ is near to the minimal possible value with respect to the dimension of the projective space $\mathbb{P}^r$ for which $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, say $r+9\le g\le r+11$. In the course of proofs of key results, we show the existence of linearly normal curves of degree $d\ge g+1$ with arbitrarily given index of speciality with some mild restriction on the genus $g$.
Submission history
From: Changho Keem [view email][v1] Sun, 3 Dec 2023 10:57:49 GMT (79kb)
[v2] Wed, 7 Feb 2024 16:40:28 GMT (82kb)
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