Conferencia en la Universidad de  Michigan, el 2 de septiembre de 2015. Invitada por M. Mustata.

Mather discrepancy as an embedded dimension in the space of arcs

The space of arcs X_\infty of a singular variety X over a perfect field k has finiteness properties when we localize at its stable points. This allows to associate or recover invariants of X from its space of arcs. In the talk I will show some general properties of the stable points, pointing out our interest in computing the dimension of the complete local ring \^O_{X_\infty,P_E} when P_E is the stable point defined by a divisorial valuation v_E on X. I will also present our last result, together with H. Mourtada: assuming char k = 0, we have \embdim(\^O_{X_\infty,P_E}) = \^k_E + 1 where \^k_E is the Mather discrepancy of X with respect to ν_E. Expressed in terms of cylinders, stable points are precisely the generic points of the irreducible cylinders in X_\infty, and our result with H. Mourtada asserts that the embedding dimension of \^O_{X_\infty,P_E} is equal to the codimesion as cylinder of N_E, being N_E the closure of P_E in X_\infty. But in general we have \dim(\^O_{X_\infty,P_E}) < \embdim(\^O_{X_\infty,P_E}).