IDEALS GENERATED BY 2-MINORS: BINOMIAL EDGE IDEALS AND POLYOMINO IDEALS

Cycle 32th Oral Defence of the Phd Thesis

11 February 2020
Versione stampabile

Venue: Povo Zero, via Sommarive 14 (Povo) – Seminar Room “-1”
Time: 10.30 a.m.

  • Carla Mascia - PhD in Mathematics

Abstract:
Since the early 1990s, a classical object in commutative algebra has been the study of binomial ideals. A widely-investigated class of binomial ideals is the one containing those generated by a subset of 2-minors of an (m x n)-matrix of indeterminates. This presentation is devoted to illustrate some algebraic and homological properties of two classes of ideals of 2-minors: binomial edge ideals and polyomino ideals.
Binomial edge ideals arise from finite graphs and their appeal results from the fact that their homological properties reflect nicely the combinatorics of the underlying graph. We give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs.
Polyomino ideals arise from polyominoes, plane figures formed by joining one or more equal squares edge to edge. We give a necessary condition for the primality of polyomino ideals with respect to the geometric representation of the polyomino. This condition is related to a sequence of inner intervals contained in the polyomino, called a zig-zag walk, whose existence determines the non-primality of the polyomino ideal. Moreover, we present an infinite class of prime polyomino ideals.

Supervisors: Massimiliano Sala - Giancarlo Rinaldo