Given an algebraic variety $X$ with a point $x \in X$, the multiplicity of $X$ at $x$ is defined as the multiplicity of the maximal ideal of $x$ in the local ring $\mathcal{O}_{X,x}$.
In https://www.encyclopediaofmath.org/index.php/Multiplicity_of_a_singular_point this notion is defined and a few properties are given. From this site i quote: "The multiplicity does not change when $X$ is cut by a generic hypersurface through $x$".
(I guess here we should assume $X \subset \mathbb{P}^n$)
My question is now, what are exactly the conditions so that it does change? And how to prove the claim?
Any reference to the literature or just a sketch of a proof is already fine! Thanks!