Enumeration of complex and real surfaces via tropical geometry

Hannah Markwig, Thomas Markwig*, Eugenii Shustin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We prove a correspondence theorem for singular tropical surfaces in 3, 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 3. As application we show that there are pencils of real surfaces of degree d in 3 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].

Original languageEnglish
Pages (from-to)69-100
Number of pages32
JournalAdvances in Geometry
Issue number1
StatePublished - 1 Jan 2018


  • Tropical geometry
  • discriminants
  • real singular surfaces
  • tropical singular surface


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