Toric ideals of graphs and some of their homological invariants

Abstract: The study of toric ideals of graphs lies in the intersection of commutative algebra, algebraic geometry, and combinatorics. Formally, if $G = (V,E)$ is a finite simple graph with edge set $E =\{e_1,\ldots,e_s\}$ and vertex set $V = \{x_1,\ldots,x_n\},$ then the toric ideal of $G$ is the kernel of the ring homomorphism $\varphi:k[e_1,\ldots,e_s] \rightarrow k[x_1,\ldots,x_n]$ where $\varphi(e_i) = x_jx_k$ if the edge $e_i = \{x_j,x_k\}$. Ideally, one would like to understand how the homological invariants (e.g. graded Betti numbers) of $I_G$ are related to the graph $G$. In this talk I will survey some results connected to this theme, with an emphasis on the Castelnuovo-Mumford regularity of these ideals.

commutative algebra

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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