ARC SCHEME AND HIGHER DIFFERENTIAL FORMS

Let k be a field. In this article, we identify the component of weight 2 of the natural Gm,kgraduation on the k-algebra of the arc scheme attached to an affine algebraic variety X with the module of the 2-nd order derivations on X. We in particular deduce, from this property, characterizations of the geometry of hypersurfaces (in affine spaces) in terms of the nilpotency on arc scheme.