On Semicontinuity of Multiplicities in Families

  • Ilya Smirnov

    Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden
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This paper investigates the behavior of Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity in families of ideals. We show that Hilbert-Samuel multiplicity is upper semicontinuous and that Hilbert-Kunz multiplicity is upper semicontinuous in families of finite type. As a consequence, F-rational signature, an invariant defined by Hochster and Yao as the infimum of relative Hilbert-Kunz multiplicities, is, in fact, a minimum. This gives a different proof for its main property: F-rational signature is positive if and only if the ring is F-rational. The tools developed in this paper can be also applied to families over Z\mathbb{Z} and yield a solution to Claudia Miller's question on reduction mod pp of Hilbert-Kunz function.

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Ilya Smirnov, On Semicontinuity of Multiplicities in Families. Doc. Math. 25 (2020), pp. 381–399

DOI 10.4171/DM/751