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Enumeration of complex and real surfaces via tropical geometry

  • Hannah Markwig , Thomas Markwig EMAIL logo and Eugenii Shustin
Published/Copyright: June 9, 2017
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Advances in Geometry
From the journal Advances in Geometry Volume 18 Issue 1

Abstract

We prove a correspondence theorem for singular tropical surfaces in ℝ3, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we develop a three-dimensional version of Mikhalkin’s lattice path algorithm that enumerates singular tropical surfaces passing through an appropriate configuration of points in ℝ3. As application we show that there are pencils of real surfaces of degree d in ℙ3 containing at least (3/2)d3 + O(d2) singular surfaces, which is asymptotically comparable to the number 4(d − 1)3 of all complex singular surfaces in the pencil. Our result relies on the classification of singular tropical surfaces [12].

Keywords: Tropical geometry; tropical singular surface; discriminants; real singular surfaces
MSC 2010: 14T05; 51M20; 14N10

Communicated by: C. Scheiderer

  1. Funding: The research was supported by the German-Israeli Foundation grant no. 1174-197.6/2011, by the Minerva–Minkowski Center for Geometry at the Tel Aviv University and by the DFG-grant MA 4797/5-1.

Acknowledgements

A substantial part of this work has been done during the authors’ visit to the Centre Interfacultaire Bernoulli, Lausanne, and during the third author’s visit to the Institut des Hautes Études Scientifiques. We are very grateful to these institutions for the hospitality and excellent working conditions. The authors would like to thank the unknown referee for valuable remarks and suggestions, which helped us to improve the presentation.

References

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Received: 2015-7-14
Revised: 2016-4-4
Published Online: 2017-6-9
Published in Print: 2018-1-26

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  3. The non-solvable doubly transitive dimensional dual hyperovals
  5. On the real differential of a slice regular function
  7. Generalized null 2-type immersions in Euclidean space
  9. A dual rigidity of the sphere and the hyperbolic plane
  11. The conjugacy locus of Cayley–Salmon lines
  13. Steiner’s Porism in finite Miquelian Möbius planes
  15. Enumeration of complex and real surfaces via tropical geometry
  17. On complete Yamabe solitons
  19. Polytopal approximation of elongated convex bodies
  21. A note on commutative semifield planes
  23. Quartic surfaces with icosahedral symmetry
https://doi.org/10.1515/advgeom-2017-0024
Keywords for this article
Tropical geometry; tropical singular surface; discriminants; real singular surfaces

Articles in the same Issue

  1. Frontmatter
  3. The non-solvable doubly transitive dimensional dual hyperovals
  5. On the real differential of a slice regular function
  7. Generalized null 2-type immersions in Euclidean space
  9. A dual rigidity of the sphere and the hyperbolic plane
  11. The conjugacy locus of Cayley–Salmon lines
  13. Steiner’s Porism in finite Miquelian Möbius planes
  15. Enumeration of complex and real surfaces via tropical geometry
  17. On complete Yamabe solitons
  19. Polytopal approximation of elongated convex bodies
  21. A note on commutative semifield planes
  23. Quartic surfaces with icosahedral symmetry
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