Local tropicalizations of splice type surface singularities

TL;DR

This paper explores the tropical geometry of splice type surface singularities, characterizing their local tropicalizations, proving their Newton non-degeneracy, and providing new insights into their resolution and reconstruction from tropical data.

Contribution

It introduces a tropical perspective on splice type surface singularities, characterizes their local tropicalizations, and proves their Newton non-degeneracy, offering new proofs and reconstruction methods.

Findings

Local tropicalizations are cones over embedded splice diagrams.

Splice type singularities are Newton non-degenerate complete intersections.

Tropical methods can reprove resolution results for plane curve singularities.

Abstract

Splice type surface singularities were introduced by Neumann and Wahl as a generalization of the class of Pham-Brieskorn-Hamm complete intersections of dimension two. Their construction depends on a weighted tree called a splice diagram. In this paper, we study these singularities from the tropical viewpoint. We characterize their local tropicalizations as the cones over the appropriately embedded associated splice diagrams. As a corollary, we reprove some of Neumann and Wahl's earlier results on these singularities by purely tropical methods, and show that splice type surface singularities are Newton non-degenerate complete intersections in the sense of Khovanskii. We also confirm that under suitable coprimality conditions on its weights, the diagram can be uniquely recovered from the local tropicalization. As a corollary of the Newton non-degeneracy property, we obtain an alternative proof of a recent theorem of de Felipe, Gonz\'alez P\'erez and Mourtada, stating that embedded resolutions of any plane curve singularity can be achieved by a single toric morphism, after re-embedding the ambient smooth surface germ in a higher-dimensional smooth space. The paper ends with an appendix by Jonathan Wahl, proving a criterion of regularity of a sequence in a ring of convergent power series, given the regularity of an associated sequence of initial forms.