For m ∈ N, we give formulas for the number N (m) of irreducible components of the m-th Jet Scheme of a complex branch C and for their codimensions, in terms of m and the generators of the semigroup of C. This structure of the Jet Schemes determines and is determined by the topological type of C.

Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations we obtain a new approach to the classical Rogers-Ramanujan Identities. The linking object is the Hilbert-Poincaré series of the arc space over a point of the base variety. In the case of the double point this is precisely the generating series for the integer partitions without equal or consecutive parts. * FWF project P21461 and I382. † Postdoctoral Fellow of the Research Foundation -Flanders (FWO) MSC2010 Subject Classification: 14B05, 11P84, 05A17, 13P10.where J 0 ∞ (X) j denotes the jth homogeneous component of J 0 ∞ (X). In the simple case of X = A 1 we may use the generating function for partitions to represent HP J 0 ∞ (A 1 ) (t) by H := i≥1

We prove that for m ∈ N, m big enough, the number of irreducble components of the schemes of m−jets centered at a point which is a double point singularity is independent of m and is equal to the number of exceptional curves on the minimal resolution of the singularity. We also relate some irreducible components of the jet schemes of an E 6 singularity to its "minimal" embedded resolutions of singularities.

Abstract. Let X be a variety over a field k and let X ∞ be its space of arcs. We study the embedding dimension of the complete local ring A := O X∞,PE where P E is the stable point defined by a divisorial valuation ν E on X. Assuming char k = 0, we prove that embdim A = k E + 1 where k E is the Mather discrepancy of X with respect to ν E . We also obtain that dim A has as lower bound the MatherJacobian log-discrepancy of X with respect to ν E . For X normal and complete intersection, we prove as a consequence that points P E of codimension one in X ∞ have discrepancy k E ≤ 0.

Using the theory of jet schemes, we give a new approach to the description of a minimal generating sequence of a divisorial valuations on A 2 . For this purpose, we show how one can recover the approximate roots of an analytically irreducible plane curve from the equations of its jet schemes. As an application, for a given divisorial valuation v centered at the origin of A 2 , we construct an algebraic embedding A 2 → A N , N ≥ 2 such that v is the trace of a monomial valuation on A N . We explain how results in this direction give a constructive approach to a conjecture of Teissier on resolution of singularities by one toric morphism. v W (h) = min γ∈W {ord t γ * (h)},