Towards the Singular Locus of the Space of Arcs

We give a theory on the space of arcs X∞ of a singular variety X, based on the finiteness property of the stable points of X∞. We also show how computations can be made in this theory. As a consequence, we obtain a process of determining invariants of X from its space of arcs X∞, and from its spaces of wedges (arcs in the space of arcs). More precisely, we give two concepts of regularity for the stable points of X∞, and we show that we have regularity in codimension 1, and for all stable points of the space of arcs of a curve. As another application, we show concrete examples, obtained from the spaces of arcs, of Noetherian 1-dimensional local rings which are analytically ramified.