Narrow Results

Let $k$ be a field. We introduce a new geometric invariant, namely the minimal formal models, associated with every curve singularity (defined over $k$). This is a noetherian affine adic formal $k$-scheme, defined by using the formal neighborhood in the associated arc scheme of a primitive $k$-parametrization. For the plane curve $A_{2n}$-singularity, we show that this invariant is $\mathrm{Spf}(k[[Z]]/〈Z^{n+1}〉)$. 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.

Normalization is a fundamental ring-theoretic operation; geometrically it resolves singularities in codimension one. Existing algorithmic methods for computing the normalization rely on a common recipe: successively enlarge the given ring in form of an endomorphism ring of a certain (fractional) ideal until the process becomes stationary. While Vasconcelos' method uses the dual Jacobian ideal, Grauert–Remmert-type algorithms rely on so-called test ideals. For algebraic varieties, one can apply such normalization algorithms globally, locally, or formal analytically at all points of the variety. In this paper, we relate the number of iterations for global Grauert–Remmert-type normalization algorithms to that of its local descendants. We complement our results by a study of ADE singularities. All intermediate singularities occurring in the normalization process are determined explicitly. Besides ADE singularities the process yields simple space curve singularities from the list of Frühbis-Krüger.