Georgia Institute of Technology College of Sciences

A theorem of Bertini says that an irreducible algebraic variety remains irreducible after intersecting with a generic hyperplane.  We will discuss toric Bertini theorems for intersections with generic algebraic subtori (defined by generic binomial equations) instead of hyperplanes. As an application, we obtain a tropical Bertini theorem and a strengthening of the Structure Theorem of tropical algebraic geometry, by showing that irreducible tropical varieties remain connected through codimension one even after removing some facets.  As part of the proof of the Toric Bertini over prime characteristics, we constructed a new algebraically closed field containing the multivariate rational functions, which is smaller than previously known constructions.  This is based on joint works with Diane Maclagan, Francesca Gandini, Milena Hering, Fatemeh Mohammadi, Jenna Rajchgot, and Ashley Wheeler.