In the last decades the connections between commutative algebra and combinatorics have been extensively explored. It is interesting to study classes of ideals in a polynomial ring associating with them combinatorial objects, such as simplicial complexes, graphs, clutters or polytopes.
In this talk we are interested in the so-called binomial edge ideals, which are ideals generated by binomials corresponding to the edges of a finite simple graph G. They can be viewed as a generalization of the ideal of 2-minors of a generic matrix with two rows.
In particular, we provide a classification of Cohen-Macaulay binomial edge ideals of bipartite graphs, giving an explicit construction in graph-theoretical terms.
To prove this classification we make use of the dual graph of an ideal, showing in our setting the converse of Hartshorne's Connectedness Theorem.