Uniform Convergence Methods in Hilbert-Kunz Theory
Smirnov, Ilya, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Huneke, Craig, Department of Mathematics, University of Virginia
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.
PHD (Doctor of Philosophy)
commutative algebra, Hilbert-Kunz multiplicity, tight closure
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