Equimultiplicity in Hilbert–Kunz theory
Received: 2 November 2017 / Accepted: 12 March 2018
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract We study further the properties of Hilbert–Kunz multiplicity as a measure of
singularity. This paper develops a theory of equimultiplicity for Hilbert–Kunz multiplicity
and applies it to study the behavior of Hilbert–Kunz multiplicity on the Brenner–Monsky
hypersurface. A number of applications follows, in particular we show that Hilbert–Kunz
multiplicity attains inﬁnitely many values and that equimultiple strata may not be locally
Keywords Hilbert–Kunz multiplicity · Tight closure · Equimultiplicity
Mathematics Subject Classiﬁcation 13D40 · 13A35 · 13H15 · 14B05
Hilbert–Samuel multiplicity is a classical invariant of a local ring that generalizes the notion
of the multiplicity of a curve at a point. The multiplicity may be regarded as a measure of
singularity, where the lowest possible value, 1, corresponds to a smooth point. In study of
singularities, we are naturally led to study equimultiple points, i.e., a point such that the
Hilbert–Samuel multiplicity is constant on the subvariety deﬁned as the closure of the point.
For example, it can be considered as the weakest form of equisingularity, where we would
say that two points are equally singular if the multiplicities are equal. This notion has been
studied extensively, partially due to its appearance in Hironaka’s work on the resolution of
singularities in characteristic zero.
In 1983 Monsky deﬁned a new version of multiplicity, speciﬁc for positive characteristic.
It was called Hilbert–Kunz multiplicity, in honor of Ernst Kunz who in 1969 initiated the
Dedicated to Craig Huneke on the occasion of his 65th Birthday.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA