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Title: Local tropicalizations of splice type surface singularities
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, González Pérez 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.  more » « less
Award ID(s):
1954163 1700194
PAR ID:
10531483
Author(s) / Creator(s):
; ; ;
Publisher / Repository:
Springer-Verlag
Date Published:
Journal Name:
Mathematische Annalen
ISSN:
0025-5831
Subject(s) / Keyword(s):
Mathematics 14B05, 14T90, 32S05 (primary), 14M25, 57M15 (secondary) Surface singularities, complete intersection singularities, tropical geometry, Newton non-degeneracy.
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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