Frobenius Splittings for the toric ideals of graphs and ladder determinantal ideals

Abstract

Let F be a field and let R = F[x1, . . . , xn] be a polynomial ring. Given a polynomial f ∈ R with a squarefree initial ideal (for some monomial order), one can build a class of ideals in R call the Knutson ideals associated to f. Each Knutson ideal is radical and the set of all Knutson ideals associated to f ∈ R is closed under summation, intersection, and saturation. Each Knutson ideal Gr ̈obner degenerates to a squarefree monomial ideal. The goal of this thesis is to prove that certain classes of ideals are Knutson. The classes we focus on are toric ideals of graphs. We prove that toric ideals of certain classes of graphs are Knutson. We also show that if the toric ideal of a graph G is Knutson, and H is obtained from G by gluing an even cycle to an edge of G, then the toric ideal of H is Knutson. We also discuss the one-sided ladder determinantal ideals and prove that every one-sided ladder determinantal ideal is Knutson. In the last chapter, we discuss some future directions.