Tropical linear incidence geometry, Lorentzian polynomials, and linear series

Abstract:

Linear incidence geometry concerns the existence and enumeration of linear subspaces satisfying given incidence conditions. Over a field, linear incidence geometry is well understood. Examples include Euclid’s first postulate, Pappus’s theorem, and the whole study of Schubert calculus.

Linear spaces in the tropical world, however, can behave in unexpected ways. We will unravel some the subtlety of the incidence geometry of tropical linear spaces, give counterexamples to certain incidence properties expected from the classical setting, and discuss conditions on when they behave as expected. Then, we will introduce a class of moduli spaces, called the relative Dressians, that are useful in understanding the geometry of tropical linear spaces. Finally, we will explain the implications of tropical linear incidence geometry to the following three recent developments: Lorentzian polynomials, linear series on tropical curves, and the geometry of Grassmannians over hyperfields.