Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with index of speciality five and beyond

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 $g$.

Figures (2)

• Figure 1: Existence and non-existence; $\alpha=g-d+r=5$
• Figure 2: Irreducibility map

Theorems & Definitions (56)

• Remark 1.1
• Proposition 1.2
• Definition 2.1
• Remark 2.2
• Remark 2.3
• Proposition 2.4: Harris
• Proposition 2.5
• Remark 2.6
• Lemma 2.7
• Remark 2.8: Castelnuovo-Severi inequality