If $x \leadsto x'$ is a nontrivial specialization of points of $X$, then

$\dim _ x(X) \leq \dim _{x'}(X)$,

$\dim (\mathcal{O}_{X, x}) < \dim (\mathcal{O}_{X, x'})$,

$\text{trdeg}_ k(\kappa (x)) > \text{trdeg}_ k(\kappa (x'))$, and

any maximal chain of nontrivial specializations $x = x_0 \leadsto x_1 \leadsto \ldots \leadsto x_ n = x$ has length $n = \text{trdeg}_ k(\kappa (x)) - \text{trdeg}_ k(\kappa (x'))$.

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