On the multiplicities of an ideal on a smooth variety 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?
 A: 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:

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*O. Villamayor U, An introduction to constructive desingularization, 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

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*A. Bravo, S. Encinas, O. Villamayor, A Simplified Proof of Desingularization and Applications, Rev. Mat. Iberoam. 21 (2005), no. 2, pp. 349–458, doi: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.
