A filtration defined by arcs on a variety

Abstract

We define a natural filtration on the ring O V,0 of germs of functions on a germ of a complex analytic variety (V, 0) related with the geometry of arcs on the variety and describe some properties of it.Let (V, 0) be a germ of a complex analytic variety and let O V,0 be the ring of germs of functions on it. There is a natural filtration on the ring O V,0 defined by powers of the maximal ideal. We define another natural filtration related with the geometry of arcs on (V, 0) and describe some properties of it.An arc φ on (V, 0) is a germ of a complex analytic mapping φ : (C, 0) → (V, 0). For a function g ∈ O V,0 , its order v φ (g) on the arc φ is defined as the order of the composition g • φ, i.e., as the power of the first non-vanishing term in the power series expansion g • φ(τ ) = aτ v φ (g) + . . ., a = 0 (if g • φ ≡ 0, then v φ (g) is assumed to be equal to +∞). Let v(g) := min {φ} v φ (g). One has v(g) < ∞ for g = 0. One can easily see that v : O V,0 \ {0} → Z ≥0 is a valuation.