Binomial edge ideals of bipartite graphs

Binomial edge ideals are ideals generated by binomials corresponding to the edges of a graph, naturally generalizing the ideals of 2-minors of a generic matrix with two rows. They also arise in Algebraic Statistics in the context of conditional independence ideals. We give a combinatorial classification of Cohen-Macaulay binomial edge ideals of bipartite graphs providing an explicit construction in graph-theoretical terms. In the proof we use the dual graph of an ideal, showing in our setting the converse of Hartshorne’s Connectedness theorem. As a consequence, we prove for these ideals a Hirsch-type conjecture of Benedetti-Varbaro.

This is a joint work with Davide Bolognini and Francesco Strazzanti.