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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?

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I've fixed the Latex. –  Martin Brandenburg Mar 22 '12 at 13:10
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up vote 6 down vote accepted

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:




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.

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