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

TL;DR

This paper proves that the Hilbert scheme components of smooth curves in projective space are not rigid in moduli for certain parameters, implying only twisted cubics in P^3 are rigid under projective transformations.

Contribution

It demonstrates the non-existence of moduli-rigid components in the Hilbert scheme for curves in P^3 and certain higher-dimensional projective spaces, extending known irreducibility results.

Findings

No components rigid in moduli for g > 0, r=3.

Only twisted cubic curves in P^3 are rigid in moduli.

Irreducibility of al{H}_{d,g,3} beyond previous known ranges.

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 $\pi:\mathcal{H}_{d,g,r} \dashrightarrow \mathcal{M}_{g}$ 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 $r \geq 4$, 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.