Topics in Singularities and Jet Schemes.

Abstract

Jet schemes and arc spaces play important roles in the study of singularities of higher dimensional algebraic varieties. In this dissertation, we prove results in two directions. The first one is concerned with the study of singularities of Brill-Noether loci, in particular, the theta divisor, in terms of their jet schemes. The other concerns the generalization of the correspondence between divisorial valuations and closed irreducible cylinders in the arc space to arbitrary characteristics.