A toric variety is an algebraic variety containing the algebraic torus (C*)n as an open dense subset such that the action of the torus extends to the whole variety. Every n-dimensional toric variety can be associated to a fan, which can be given as a set of primitive vectors in an n-dimensional lattice N. Because of this association, many properties of toric varieties can be studied using combinatorial methods. This group was interested in determining which projective toric varieties are Frobenius split. If a projective toric variety is Frobenius split, it has many nice properties. For example, it can be given as the solution set to homogeneous degree two polynomials.
Sam Payne proved in 2008 that a toric variety is Frobenius split if and only if an associated polytope called the splitting polytope contains representatives of every residue class of (1/q)M/M where M is the dual lattice to N. The group’s goal was to use this characterization of Frobenius splitting to classify the Frobenius split projective toric varieties in n dimensions. Given a trivalent tree where edges are labeled with variables and integers, it’s possible to construct the fan of a toric variety. They determined which toric varieties arising in this way are Frobenius split for certain classes of edge labelings.