A remark on Berger's conjecture, Kolchin's theorem, and arc schemes (Q515522)

This note relates the so-called Berger conjecture for curves to properties of arc schemes. For a curve \(C\) over a perfect field \(k\) and a point \(P\) of \(C\), let \(\mathcal O_P\) denote its local ring at \(P\). Berger's conjecture states that \(C\) is smooth at \(P\) if and only if the \(\mathcal O_P\)-module of \(1\)-forms \(\Omega^1_{\mathcal O_P/k}\) on \(C\) is torsion free. Inspired by results of Mustaţǎ, relating smoothness to properties of jet schemes, the author proves as main result, in arbitrary dimension, the following link between properties of arc schemes and torsion of modules of \(1\)-forms. Assume that \(k\) is of characteristic zero, and let \(V\) be an integral \(k\)-variety. If the arc scheme of \(V\) is reduced, then the \(\mathcal O_V\)-module \(\Omega^1_{V/k}\) is torsion free. The proof is based on Kolchin's Irreducibility Theorem in differential algebra.