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arXiv · 1902.07460

On semicontinuity of multiplicities in families

Ilya Smirnov
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Abstract

The paper investigates the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities in families of ideals. It is shown that Hilbert-Samuel multiplicity is upper semicontinuous almost generally and that Hilbert-Kunz multiplicity is upper semicontinuous in families of finite type. Surprisingly, our machinery can be applied for families over Z and yields a partial solution to the question about characteristic zero Hilbert-Kunz multiplicity posed by Brenner, Li, and Miller. Another application is that for an affine ring the infimum in the definition of F-rational signature, an invariant defined by Hochster and Yao, is attained.

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Ilya Smirnov. 2019-02-20. On semicontinuity of multiplicities in families. https://arxiv.org/abs/1902.07460

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