Algebra and Geometry Seminar
The arc space of a variety is an infinite dimensional scheme whose geometric structure captures, in a way that is not yet fully understood, certain features of the singularities of the variety. Focusing on its local rings and invariants of these rings such as embedding dimension and codimension, we explore the local structure of arc spaces. Our main tools rely on a formula for the sheaf of differentials on arc spaces and some recent finiteness results on the fibers of the map induced at the level of arc spaces from an arbitrary morphism of schemes over a field. The talk is based on joint work with Christopher Chiu and Roi Docampo.