Abstract
Abstract We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure of singularities, our main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. As an application, we complement a theorem of Drinfeld, Grinberg and Kazhdan on formal neighbourhoods in arc spaces by providing a converse to their theorem, an optimal bound for the embedding codimension of the formal model appearing in the statement, a precise formula for the embedding dimension of the model constructed in Drinfeld’s proof and a geometric meaningful way of realising the decomposition stated in the theorem.
Highlights
We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties and study in detail the case of arc spaces of schemes of finite type over a field
We extend the definition of the embedding codimension to arbitrary local rings ( A, m, k) by setting ecodim( A) := ht(ker(γ)), where γ : Symk (m/m2) → gr( A) is the natural homomorphism
For every equicharacteristic local ring ( A, m, k), we have ecodim( A) ≤ fcodim( A), and equality holds in the following cases: 1. the ring A has embedding dimension edim( A) < ∞ or 2. there exists a scheme X of finite type over k such that A is isomorphic to the local ring of the arc space of X at a k-rational point
Summary
In this paper we will work with rings of power series in an arbitrary number of indeterminates. A formal embedding τ is called efficient if the induced map at the level of continuous Zariski cotangent spaces n/n2 → m/m2 is an isomorphism. Proposition 3.10 implies the existence of formal coefficient fields for any equicharacteristic local ring ( A, m, k). We finish this section by recalling the following result, which guarantees the existence of a Zariskilocal minimal embedding for singular points of a scheme of finite type over an infinite field. This is well known in the case of complex varieties (see, for example, [6, Theorem 3]) and we provide an extension of the proof to the more general case considered here.
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