We study initial algebras of determinantal rings, defined by minors of generic matrices, with respect to their classical generic point. This approach leads to very short proofs for the structural properties of determinantal rings. Moreover, it allows us to classify their Cohen-Macaulay and Ulrich ideals.

Let A be a Noetherian local ring with the maximal ideal m and d = dim A. Let Q be a parameter ideal in A. Let I = Q : m. The problem of when the equality I = QI holds true is explored. When A is a Cohen-Macaulay ring, this problem was completely solved by A. Corso, C. Huneke, C. Polini, and W. Vasconcelos [CHV, CP, CPV], while nothing is known when A is not a Cohen-Macaulay ring. The present pu...