Irreducibility and components rigid in moduli of the Hilbert scheme of smooth curves

Abstract

Denote by ${\mathcal{H}}_{d,g,r}$ the Hilbert scheme of smooth curves, that is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in ${\mathbb{P}}^r$. A component of ${\mathcal{H}}_{d,g,r}$ is rigid in moduli if its image under the natural map $\pi :{\mathcal{H}}_{d,g,r}⇢{\mathcal{M}}_g$ is a one point set. In this note, we provide a proof of the fact that ${\mathcal{H}}_{d,g,r}$ has no components rigid in moduli for $g>0$ and $r=3$, from which it follows that the only smooth projective curves embedded in ${\mathbb{P}}^3$ whose only deformations are given by projective transformations are the twisted cubic curves. In case $r\ge 4$, we also prove the non-existence of a component of ${\mathcal{H}}_{d,g,r}$ rigid in moduli in a certain restricted range of $d$, $g>0$ and $r$. In the course of the proofs, we establish the irreducibility of ${\mathcal{H}}_{d,g,3}$ beyond the range which has been known before.

arXiv commentv2: added (ii) in Cor. 3.6; v3: minor typos corrected