Towards tropically counting binodal surfaces

Based on Mikhalkin's Correspondence Theorem, tropical geometry has been successfully used to count plane curves satisfying point conditions. For curves there exist nice tools using tropical geometry to break the task of counting curves down to a combinatorial problem. The generalization for higher dimensional varieties, like surfaces, is more involved.

After a brief introduction to tropical geometry, I will present tropical floor plans as developed by Markwig et. al. as a tool, which allows to count not only curves but also multi-nodal surfaces under certain constraints. In joint work with Madeline Brandt, we extended the definition of tropical floor plans to count even more cases of surfaces:

In particular in our newest paper, we looked at the case when two nodes are tropically close together, i.e., unseparated. We then prove that for δ=2 or 3 nodes, tropical surfaces with unseparated nodes contribute asymptotically to the second order term of the polynomial giving the degree of the family of complex projective surfaces in ℙ3 of degree d with δ nodes. We classified the cases when two nodes in a surface tropicalize to a vertex dual to a polytope with 6 lattice points, and proved that this only happens for projective degree d surfaces satisfying point conditions in Mikhalkin position when d>4.