Uniform Convergence Methods in Hilbert-Kunz Theory

Hilbert-Kunz multiplicity is an invariant of a local ring containing a field of positive characteristic. In this work, we study its continuity properties as a function on a variety.

First, we develop a theory of equimultiplicity for Hilbert-Kunz multiplicity.
Remarkably, it is quite similar to the classical equimultiplicity. The theory is then applied to show that a stronger form of upper semi-continuity does not hold.

Later, using uniform convergence ideas we prove that a weaker form of upper semi-continuity holds. As an application, we obtain that the maximum value locus of Hilbert-Kunz multiplicity is closed.