Local volumes, integral closures, and equisingularity

• Online Access: http://hdl.handle.net/2047/D20248936

In this work we introduce the local volume of a line bundle as a tool of deformation theory. One of our main results determines its change across families of schemes o finite type over a field generalizing results of Gaffney and Teissier about the Hilbert-Samuel and Buchsbaum--Rim multiplicities. We show that the local volume of a particular class of line bundles provides numerical control of Whitney--Thom equisingularity for families of a large class of isolated singularities generalizing well-known results about smoothable singularities. The new large class consists of singularities that admit deformations to deficient conormal singularities (DCS). This class contains as a subclass all smoothable singularities. For singularities that admit DCS deformations, the local volume has particularly nice behavior. Studying the properties of the local volume requires strengthening and generalizing some classical results about asymptotics of prime divisors. In the second part of this work we present a new approach to studying these problems which leads to substantial generalizations of results of Rees, Burch, McAdam and Katz.