Geodesic
On Semicontinuity of Multiplicities in Families
Documenta mathematica, Tome 25 (2020), pp. 381-399.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $\mathbb{Z}$ and yield a solution to Claudia Miller's question on reduction mod $p$ of Hilbert-Kunz function.
Classification : 13A35, 13D40, 13H15, 14B05, 14B07, 14D06
Keywords: multiplicity, Hilbert-Samuel polynomial, Hilbert-Kunz multiplicity, semicontinuity, families 
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     author = {Smirnov, Ilya},
     title = {On {Semicontinuity} of {Multiplicities} in {Families}},
     journal = {Documenta mathematica},
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     year = {2020},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a56/}
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Smirnov, Ilya. On Semicontinuity of Multiplicities in Families. Documenta mathematica, Tome 25 (2020), pp. 381-399. http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a56/