| Résume | In this talk our goal is to investigate the schematic structure of the space of arcs and in particular its local geometry at singular points. One of the main ingredients will be the formula for the sheaf of differentials of the arc space proven by de Fernex and Docampo, which we will obtain using higher derivations of modules as originally introduced by Ribenboim. We will then present two ways of generalizing the notion of embedding codimension to non-Noetherian local rings, giving a formal invariant which measures the 'size' of a singularity. Our main result is a characterization of nondegenerate arcs as those of finite embedding codimension, with an explicit bound provided in this case. We will finish by comparing our results to the Drinfeld-Grinberg-Kazhdan theorem on the formal neighborhood of nondegenerate arcs. This talk covers joint work with Luis Narvaez Macarro as well as Tommaso de Fernex and Roi Docampo. |
