Embedding codimension of the space of arcs

Abstract We introduce a notion of embedding codimension an arbitrary local ring, establish some general properties and study in detail the case arc spaces schemes finite type over field. Viewing as measure singularities, our main result can be interpreted saying that singularities space are maximal at arcs fully embedded singular locus underlying scheme, progressively improve we move away from said locus. As application, complement theorem Drinfeld, Grinberg Kazhdan on formal neighbourhoods by providing converse to their theorem, optimal bound for model appearing statement, precise formula dimension constructed Drinfeld’s proof geometric meaningful way realising decomposition stated theorem.