Special Year Seminar II
Topological Bound for Tropical Varieties
The construction by Mikhalkin of a non-planar tropical cubic curve in R^3 of genus 1 marked a significant breakthrough in the study of combinatorial tropical varieties. It was the first known example of a non-realizable tropical variety, with the obstruction being of a topological nature.
More recently, it was shown that the upper bound on the top Betti number of combinatorial tropical varieties, for a given dimension and degree, also depends on the codimension, which contrasts with the complex case. Specifically, for any three positive integers d, m, and k, there exist tropical varieties of degree d, dimension m, and codimension k whose top Betti number is equal to k times the upper bound on the top Betti number of complex varieties of degree d and dimension m. The method used to construct these examples is known as Floor composition. For lower dimensions and degrees, these constructions are maximal.
This is joint work with Benoît Bertrand and Erwan Brugallé.