Local tropicalizations of splice type surface singularities (Q6624752)

Splice diagrams are finite trees with integer weights assigned to half-edges and \(\pm\) signs to vertices. Eisenbud and Neumann used certain type of splice diagrams to study plane curve singularities [\textit{D. Eisenbud} and \textit{W. Neumann}, Three-dimensional link theory and invariants of plane curve singularities. Princeton, NJ: Princeton University Press (1985; Zbl 0628.57002)]. In later years, Neumann and Wahl introduced ``splice type surface singularities'' as a generalization of Pham-Brieskorn-Hamm complete intersection singularities [\textit{W. D. Neumann} and \textit{J. Wahl}, in: Trends in singularities. Basel: Birkhäuser. 181--190 (2002; Zbl 1072.14502)].\N\NThis paper uses local tropicalization to study splice type singularities. The authors use a creative method toward local tropicalization: they start with a splice diagram \(\Gamma\) which is appropriately embedded in \(\mathbb{R}^n\). By successive stellar subdivisions of the induced simplex in \(\mathbb{R}^n\), they remove open cones in the positive orthant avoiding the local tropicalization. In addition, they show that under some coprime conditions on the weights of half-edges in \(\Gamma\), the splice diagram can be uniquely recovered from local tropicalization.\N\NAs their first result, the authors recover and strengthen the central theorem from [\textit{W. D. Neumann} and \textit{J. Wahl}, Geom. Topol. 9, 699--755 (2005; Zbl 1087.32017)] as the following: splice type systems are Newton non-degenerate complete intersection systems of equations. Moreover, the associated splice type singularities are isolated, irreducible and not contained in any coordinate subspace of the corresponding ambient space \(\mathbb{C}^n\).\N\NAs a consequence of Newton non-degeneracy, they prove that any germ of a reduced plane curve may be resolved by one toric modification after re-embedding its ambient smooth germ of surface into a higher-dimensional germ \((\mathbb{C}^n, 0)\). This is in fact an alternative proof of the main theorem in [\textit{A. B. de Felipe} et al., Math. Ann. 387, No. 3--4, 1853--1902 (2023; Zbl 1529.14007)].