Finite formal model of toric singularities

Abstract

We study the formal neighborhoods at rational nondegenerate arcs of the arc scheme associated with a toric variety. The first main result of this article shows that these formal neighborhoods are generically constant on each Nash component of the variety. Furthermore, using our previous work, we attach to every such formal neighborhood, and in fact to every toric valuation, a minimal formal model (in the class of stable isomorphisms) which can be interpreted as a measure of the singularities of the base-variety. As a second main statement, for a large class of toric valuations, we compute the dimension and the embedding dimension of such minimal formal models, and we relate the latter to the Mather discrepancy. The class includes the strongly essential valuations, that is to say those the center of which is a divisor in the exceptional locus of every resolution of singularities of the variety. We also obtain a similar result for monomial curves.

Embedding codimension of the space of arcs

“…then we can denote by Z min 𝑣 the minimal formal model of 𝑋 ∞ at any arc 𝛼 ∈ 𝑋 • ∞,𝑣 . The next theorem is one of the main results of [13]. A similar, more general property is proved for elements v satisfying a certain property called P 𝑣 ; we refer to the original source for the precise statement.…”

“…Starting from Section 8, we focus on the case of arc spaces, proving some technical theorems in Section 8 and then addressing the theorem of Drinfeld, Grinberg and Kazhdan in Sections 9 and 10. The last section is devoted to some applications related to Mather-Jacobian discrepancies, among others to the case of toric singularities using results of [13].…”

“…In their recent article [13], Bourqui and Sebag study DGK decompositions of 𝑋 ∞ at arcs that are not fully contained in the T-invariant divisor of X. The focus is on the open set 𝑋 • ∞ ⊂ 𝑋 ∞ consisting of those arcs whose generic point is in T. They prove that for any 𝛼 ∈ 𝑋 • ∞ , the local ring O 𝑋 ∞ , 𝛼 depends only on the associated valuation ord 𝛼 , and in particular so does the minimal formal model [13,Corollary 3.3]. In particular, if we set…”

“…An extension of the theorem to formal schemes is given in [10]. Formal models of toric singularities are studied in [13].…”

Deformations of arcs and comparison of formal neighborhoods for a curve singularity

“…Their work was motivated by geometric representation theory and Langlands program; subsequent works in this direction include [BNS16, Bou20, Ngô17]. On the other hand, the first named author of the present paper and Julien Sebag suggested that there should be strong connections between the geometric properties of the finite formal models of a rational arc and the nature of the singularity at the origin of the arc, and gave some first evidences for that ([BS17a, BS17c, BS17b, BS19], see also [BS20] and [Bou21]).…”