# On semicontinuity of multiplicities in families

## TL;DR

This paper studies the semicontinuity properties of Hilbert-Samuel and Hilbert-Kunz multiplicities in algebraic families, providing new insights and partial solutions to longstanding questions in commutative algebra.

## Contribution

It establishes upper semicontinuity of these multiplicities in broad settings and applies the machinery to characteristic zero cases, advancing understanding of multiplicity invariants.

## Key findings

- Hilbert-Samuel multiplicity is upper semicontinuous in general.
- Hilbert-Kunz multiplicity is upper semicontinuous in finite type families.
- The machinery applies to families over Z, partially addressing characteristic zero questions.

## 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.