Arc Schemes

Abstract

This chapter is devoted to the study of the arc schemes associated with schemes X defined over arbitrary base schemes S. Informally speaking, an arc on a scheme X is a formal germ of a curve on X, and the arc scheme $\mathcal{L}_{\infty }(X/S)$ parameterizes the arcs in the fibers of X → S. The arc scheme was originally defined by Nash (1995) to obtain information about the structure of algebraic singularities and their resolutions. It also takes the spotlight in the theory of motivic integration, as the measure space over which functions are integrated. In section 2, we construct the spaces of jets, which are approximate arcs up to finite order. The construction consists of a process of restriction of scalars à la Weil, presented in section 1. We then explain in section 3 why arc schemes exist and how to recover them as limits of jet schemes. We study their topology in section 4 and their differential properties in section 3. Finally, in section 5 we explain a local structure theorem for arc schemes due to Grinberg and Kazhdan (2000) and Drinfeld (2002).