A remark on Berger’s conjecture, Kolchin’s theorem, and arc schemes

Abstract

Let k be a field of characteristic zero. Let V be a k-scheme of finite type, i.e., a k-variety, which is integral. We prove that if the associated arc scheme ${\mathcal{L}_{\infty}(V)}$ is reduced, then the ${\mathcal{O}_{V}}$-Module ${\Omega_{V/k}^{1}}$ is torsion-free. Then if the k-variety V is assumed to be locally a complete intersection (lci), we deduce that the k-variety V is normal. We also obtain the following consequence: for every class ${\mathfrak{C}}$ of integral k-curves which satisfies the Berger conjecture, and for every ${\mathscr{C} \in \mathfrak{C}}$, the k-curve ${\mathscr{C}}$ is smooth if and only if ${\mathcal{L}(\mathscr{C})}$ is reduced.