Algebraic properties of the binomial edge ideal of a complete bipartite graph

Let J_G denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials x_iy_j − x_jy_i, 1 ≤ i < j≤ n, in the polynomial ring S = K[x₁, . . . , x_n, y₁, . . . , y_n] where {i, j} is an edge of G. We study the arithmetic properties of S/J_G for G, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of deficiencies, the duals of local cohomology modules, and prove the purity of the minimal free resolution of S/J_G.