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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.
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The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof
Splice type surface singularities, introduced in 2002 by Neumann and Wahl, provide all examples known so far of integral homology spheres which appear as links of complex isolated complete intersections of dimension two. They are determined, up to a form of equisingularity, by decorated trees called splice diagrams. In 2005, Neumann and Wahl formulated their Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a special kind of decomposition into pieces of the Milnor fibers of the associated singularities. These pieces are constructed from the Milnor fibers of the splice type singularities determined by the subdiagrams on both sides of the chosen edge. In this paper we give an overview of this conjecture and a detailed outline of its proof, based on techniques from tropical geometry and log geometry in the sense of Fontaine and Illusie. The crucial log geometric ingredient is the operation of rounding of a complex logarithmic space introduced in 1999 by Kato and Nakayama. It is a functorial generalization of the operation of real oriented blowup. The use of the latter to study Milnor fibrations was pioneered by A'Campo in 1975.
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- PAR ID:
- 10531490
- Editor(s):
- Papadopoulos, Athanase
- Publisher / Repository:
- Essays in Geometry - IRMA Lectures in Mathematics and Theoretical Physics - EMS Press
- Date Published:
- Volume:
- 34
- ISBN:
- 978-3-98547-024-2
- Page Range / eLocation ID:
- 629-711
- Subject(s) / Keyword(s):
- Mathematics 14B05 (Primary), 14T90, 32S05, 14M25, 57M15 (Secondary) Complete intersection singularities, integral homology spheres, Kato–Nakayama spaces, local tropicalization, log geometry, Milnor fibers, Newton non-degeneracy, real oriented blowups, rounding, Seifert fibrations, surface singularities, splice type singularities, toric geometry, toroidal varieties, tropical geometry.
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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