Let $X$ be a smooth variety, $\xi$ be a point of $X$ and $\mathfrak{a}$ be an ideal sheaf. If we define $mult\_{\xi} \mathfrak{a}$ to be the largest integer $p$ such that $\mathfrak{a} \cdot \mathcal{O}\_{X,\xi}\subseteq \mathfrak{m}\_{\xi}^{p}$, where $\mathfrak{m}\_{\xi}$ is the maximal ideal of $\mathcal{O}\_{X,\xi}$, we have the map $\xi \mapsto mult\_{\xi}\mathfrak{a}$.


Is this map upper-semicontinuous?



On the multiplicities of an ideal on a smooth variety
This is true. Indeed, this function is often called the *order of the ideal $\mathfrak{a}$ at a point $\xi$*. This function shows up quite a lot in modern proofs of resolution of singularities,


For example:


* O. Villamayor U, [An introduction to constructive desingularization](https://arxiv.org/abs/math/0507537), Journal of Symbolic Computation
Volume 39, Issues 3–4, March–April 2005, Pages 465-491, <https://doi.org/10.1016/j.jsc.2004.11.014>, arXiv:math/0507537,


or


* A. Bravo, S. Encinas, O. Villamayor, *[A Simplified Proof of Desingularization and Applications](https://arxiv.org/abs/math/0206244)*, Rev. Mat. Iberoam. 21 (2005), no. 2, pp. 349–458, doi:[10.4171/RMI/425](https://doi.org/10.4171/RMI/425), arXiv:math/0206244.


For a brief sketch of the semi-continuity property, see page 33 of the second linked paper.


The idea is that the question reduces to principal $\mathfrak{a}$, and then the order you define coincides with the ordinary multiplicity, which is semi-continuous. Indeed, there are even stronger statements, the Hilbert-Samuel function itself is semi-continuous by a result of Bennett, *On the characteristic functions of a local ring*, Ann. of Math. 91 1970 25–87 [JSTOR](https://www.jstor.org/stable/1970601).


