A major caveat is that snc models are highly non-unique, because one can always blow up a given snc model along some locus inside the central fibre. The intrinsic information concerning polarized degenerations, such as the limiting behaviour of metrics and volume measures, are independent of particular snc models.

The effect of blow up on the dual intersection complex and essential skeleton is well understood (cf. [49, A.4]). The intuitive picture is as follows:

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  If we blow up an snc model $\mathcal{X}$ along a smooth irreducible subvariety properly contained inside $E_{J}=\cap_{j\in J}E_{j}$, but not contained in any smaller intersection stratum, then the new dual intersection complex contains $\Delta_{\mathcal{X}}$, while introducing a new vertex, and some new wings over certain faces of $\Delta_{\mathcal{X}}$. The essential skeleton remains intact.

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  If we blow up an snc model $\mathcal{X}$ along some $E_{J}=\cap_{j\in J}E_{j}$, then the new dual intersection complex is a subdivision of $\Delta_{\mathcal{X}}$. If the simplex $\Delta_{J}$ corresponding to $E_{J}$ is a face of the essential skeleton $Sk(\mathcal{X})$, then there is an induced subdivision on $Sk(\mathcal{X})$, and otherwise the essential skeleton remains intact.

While $\Delta_{\mathcal{X}}$ generally becomes larger under blow ups, the essential skeleton $Sk(\mathcal{X})$ is only subdivided, and its piecewise affine structure (in particular its homeomorphism type) is a birational invariant, denoted as $Sk(X)$. The birational invariance is not surprising: the essential skeleton is the measure theoretic limit of $X_{t}$, a property independent of the choice of models.