Math. Z. https://doi.org/10.1007/s00209-018-2082-5 Mathematische Zeitschrift Ilya Smirnov 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 closed. Keywords Hilbert–Kunz multiplicity · Tight closure · Equimultiplicity Mathematics Subject Classiﬁcation 13D40 · 13A35 · 13H15 · 14B05 1 Introduction 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
Mathematische Zeitschrift – Springer Journals
Published: Jun 6, 2018
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