We construct a local invariant for resolution of singularities of two-dimensional excellent Noetherian schemes with boundary. We prove that the invariant strictly decreases at every step of the algorithm of Cossart, Jannsen and Saito. X(ν) consists of finitely many mutually disjoint components X(ν), for ν ∈ Σ max X (where Σ max X denotes the set of maximal values of the Hilbert-Samuel function on X). We use Corollaries 5.18 and 5.19 of [CJS] which state ∈ N N (4) The level sets of the Hilbert-Samuel function for an element ν ∈ N N are defined asMoreover, we put Σ X := {H X (x) | x ∈ X}, and denote by Σ max X the set of maximal elements in Σ X . Then X max = ν∈Σ max X X(ν) is called the Hilbert-Samuel locus of X. We also call it the maximal Hilbert-Samuel locus of X. stable under translation, i.e., for which C x (X) + W = C x (X) holds. This vector space is the directrix Dir x (X) of the tangent cone at x. Sometimes we also call it the directrix of J at M and write Dir M (J).There is a more intrinsic definition: C x (X) is naturally embedded in the Zariski tangent space Twhere M x is the maximal ideal of O X,x . We define the directrix as the biggest subvector space of T x (X) leaving C x (X) stable under translation.As T x (X) is a subspace of Spec(gr M (R)), both definitions coincide. See [CJS] Definition 1.26.Definition 1.6. Let J ⊂ R be the ideal which defines X locally at x.(1) A system of regular elements (y) = (y 1 , . . . , y r ) in R is said to determine the directrix Dir x (X) of X at x if the generators of In M (J) ⊂ gr M (R) are contained in k(x)[Y ], Y j = y j mod M 2 , for 1 ≤ j ≤ r, ( In M (J) ∩ k(x)[Y ] ) gr M (R) = In M (J),and, moreover, (y) is required to be minimal with this property, i.e., the number of elements r has to be minimal.

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We show that given a face of Hironaka's characteristic polyhedron, it does only depend on the singularity and a flag defined by the linear form determining the face. As a consequence we get that certain numerical data obtained from the characteristic polyhedron are invariants of the singularity. In particular, they do not depend on an embedding.

In this paper we study Hironaka's idealistic exponents in the situation over Spec (Z). In particular we give an idealistic interpretation of the tangent cone, the directrix, and the ridge. The main purpose is to introduce the notion of characteristic polyhedra of idealistic exponents and deduce from them intrinsic data on the idealistic exponent.