Twisted Hilbert schemes and division algebras

Abstract: 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.