Rationality of Hilbert-Kunz Multiplicities: A Likely Counterexample

摘要

At a 2004 Banff workshop, I gave a talk to demonstrate that, in many cases of interest, the Hilbert-Kunz multiplicity of a hypersurface is a rational number. (Mel Hochster, in the audience, told me a curious general fact: the set of possible Hilbert-Kunz multiplicities is countable.)rnAt the time I suspected that Hilbert-Kunz multiplicities must be rational. But soon after the workshop I found reason to change my opinion, and in this paper I suggest that a certain hypersurface defined by a 5-variable polynomial has 4/3 + 5/(14 7~(1/2)) as its Hilbert-Kunz multiplicity.