Inner geometry of complex surfaces: a valuative approach

On Lipschitz normally embedded complex surface germs

“…By definition, a vertex w of Γ σ is the root vertex (respectively, a Δ-node) of Γ σ if and only if −1 (w) contains an L-node (respectively a P-node) of (X, 0). To prove part (i) of the proposition it is then sufficient to establish the following claim: a vertex w of Γ σ that is not On Lipschitz normally embedded complex surface germs (v) The inner rates of the vertices of Γ σ Δ are determined by those of the vertices of Γ π because inner rates commute with thanks to [BFP22,Lemma 3.2].…”

“…By Corollary 4.6, there exist a refinement of and a sequence of point blowups such that induces a morphism of graphs . In particular, because respects inner rates (see again [BFP22, Lemma 3.2]), all edges of that are incoming at are sent to the unique edge of that is incoming at (its uniqueness can for example be seen as a consequence of Proposition 5.1). The lemma follows now by applying the formula of [BFP22, Lemma 4.18], observing that the local degree along every edge adjacent to equals 1 by Lemma 4.1(ii) and that, even if further point blowups may be needed to pass from to a resolution adapted to , no new edge can be incoming at .…”

“…Observe that, because the exceptional component of the blowup of an intersection point between the two components associated with and has multiplicity , and , the metric on is compatible with subdividing the edges of the graph by blowing up at double points of , and thus induces a metric on . The reader should be warned that this metric on is not the same as that defined in [BFP22, § 2.1], albeit it is strictly related to the latter and was already briefly used in Lemma 5.5 of [BFP22].…”

“…Denote by the unique -node of , that is, the divisorial point associated with the blowup of at . For every divisorial point of the inner rate of is by [BFP22, Lemma 5.5] (or, in a more elementary way, by a simple computation using Lemma 3.6 of [BFP22]). As the inner rates on and commute with the map (see [BFP22, Lemma 3.2]), we need to show that .…”

“…Although the first two parts of the theorem are quite elementary, the remaining parts rely heavily on a careful study of generic projections of LNE surfaces (see Lemma 4.1), building on results from [NPP20a]. Parts (iii) and (iv) also depend on the study of inner rates of [BFP22], and, in particular, on the so-called Laplacian formula of [BFP22].…”