The minimal formal models of curve singularities

Abstract

Let [Formula: see text] be a field. We introduce a new geometric invariant, namely the minimal formal models, associated with every curve singularity (defined over [Formula: see text]). This is a noetherian affine adic formal [Formula: see text]-scheme, defined by using the formal neighborhood in the associated arc scheme of a primitive [Formula: see text]-parametrization. For the plane curve [Formula: see text]-singularity, we show that this invariant is [Formula: see text]. We also obtain information on the minimal formal model of the so-called generalized cusp. We introduce various questions in the direction of the study of these minimal formal models with respect to singularity theory. Our results provide the first positive elements of answer. As a direct application of the former results, we prove that, in general, the isomorphisms satisfying the Drinfeld–Grinberg–Kazhdan theorem on the structure of the formal neighborhoods of arc schemes at non-degenerate arcs do not come from the jet levels. In some sense, this shows that the Drinfeld–Grinberg–Kazhdan theorem is not a formal consequence of the Denef–Loeser fibration lemma.