Knutson ideals

In this thesis, we begin the study of a new class of ideals, called Knutson ideals, which has several useful connections with different aspects of commutative algebra, such as $F$-singularity theory, Gr"obner basis theory, determinantal rings theory and combinatorics. First, we show that the main properties that this class has in polynomial rings over F_p are preserved when one introduces the definition of Knutson ideal also in polynomial rings over fields of characteristic zero. Then, we discuss the case of determinantal ideals of Hankel matrices and generic matrices, proving that both of them are Knutson ideals. In particular, in positive characteristic, they all define F-pure rings. In the case of Hankel matrices, we also give a characterization of all the ideals belonging to the family. Interestingly, it turns out that they are all Cohen-Macaulay ideals. In the case of generic matrices, we obtain a useful result about Gr"obner bases of certain sums of determinantal ideals. More specifically, given I = I_1 +...+ I_k a sum of ideals of minors on adjacent columns or rows, we prove that the union of the Gr"obner bases of the I_j's is a Gr"obner basis of I. Lastly, we focus on the connection between Knutson ideals and binomial edge ideals associated to weakly closed graphs. Inspired by Matsuda's work on weakly closed graphs, we show that their binomial edge ideals are Knutson ideals (in particular, they are F-pure in positive characteristic). Furthermore, we conjecture that the converse is still true, i.e, the binomial edge ideals in C_f are exactly those associated to weakly closed graphs.