We consider classes of codimension two Cohen–Macaulay ideals of a standard graded polynomial ring over a field. We revisit Vasconcelos’ problem on \(3\times 2\) matrices with homogeneous entries and, in a different angle, discuss the homological side of Geramita’s work on plane points, with additional regard to the related free resolutions and Rees algebras. An additional topic is the homological discussion of minors fixing a submatrix in the context of a perfect codimension two ideal. A combinatorial outcome of the results is a proof of the conjecture on the Jacobian ideal of a hyperplane arrangement stated by Burity, Simis and Tohǎneanu. The basic drive behind the present landscapes is a thorough analysis of the related Hilbert–Burch matrix, often without assuming equigeneration or linear presentation, not even the popular \(G_d\) condition of Artin–Nagata.
