An Introduction to the Geometry of Jet Schemes and Arc Spaces

Jet schemes and arc spaces over a variety were introduced by Nash, who was probably the first to notice that geometric properties of arcs are related to ones of the variety. Moreover, they were also discovered to play a fundamental role in the theory of motivic integration started by Kontsevich, which in turn offers powerful tools to work on them and obtain even more properties and results. 

This expository talk will be divided into two main parts. We will start defining jet schemes and arc spaces of varieties (eventually in characteristic zero), and discuss their construction as representations of certain functors. Then, we will move onto the study of their geometric properties, focusing in particular on how these provide geometric information and insight about the variety.

If time permits, we will also briefly discuss how jets and arcs are used in motivic integration, and how this powerful machinery leads to even more information.