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Towards the singular locus of the space of arcs


We give a theory on the space of arcs $X_\infty$ of a singular variety $X$, based on the finiteness property of the stable points of $X_\infty$. 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 $X_\infty$, 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 $X_\infty$, 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.