  Let $\mathscr{X}/S$ be any Severi--Brauer scheme of constant relative
dimension $n$ over a Noetherian base scheme $S$. For each polynomial
$\phi(t)\in \mathbb{Q}[t]$, we construct a scheme
$\mathrm{Hilb}_{\phi(t)}^{\mathrm{tw}}(\mathscr{X}/S)$ that \'etale locally, on
a cover $S'/S$ splitting $\mathscr{X}/S$, is the Hilbert scheme
$\mathrm{Hilb}_{\phi(t)}(\mathscr{X}_{S'}/S')$ of the projective bundle
$\mathscr{X}_{S'}/S'$.
  We then study curves of small degree on a Severi--Brauer variety in order to
analyze examples. Our primary interest, in the case $X$ is a Severi--Brauer
variety with index $n>1$ over a field $k$, is the subscheme $\mathrm{Ell}_n(X)$
of $\mathrm{Hilb}^{\mathrm{tw}}_{nt}(X/k)$ parametrizing curves that are
smooth, geometrically connected, and of genus 1.
