Diane Maclagan

Toric Bertini theorems in arbitrary characteristic

The classical Bertini theorem on irreducibility when intersecting by hyperplanes is a standard part of the algebraic geometry toolkit.  This was generalised recently, in characteristic zero, by Fuchs, Mantova, and Zannier to a toric Bertini theorem for subvarieties of an algebraic torus, with hyperplanes replaced by subtori.  I will discuss joint work with Gandini, Hering, Mohammadi, Rajchgot, Wheeler, and Yu in which we give a different proof of this theorem that removes the characteristic assumption.  The proof surprisingly hinges on better understanding algebraically closed fields containing the field of rational functions in n variables.  An application is a tropical Bertini theorem.