| Let R be a local Cohen-Macaulay ring and let I be an R-ideal. The Rees algebra R , the associated graded ring G and the fiber cone F are graded algebras that reflect various algebraic and geometric properties of the ideal I. The Cohen-Macaulay property of R and G has been extensively studied by many authors, but not much is known about the Cohen-Macaulayness of F . We give an estimate for the depth of R and G when these rings fail to be Cohen-Macaulay. We assume that I has small reduction number, sufficiently good residual intersection properties, and satisfies local conditions on the depth of some powers. We also study the Serre properties of R and G and how they are related. In particular the S 1 property for G leads to criteria for when In = I(n), where I(n ) is the n-th symbolic power of I. We prove a quite general theorem on the Cohen-Macaulayness of F that unifies and generalizes several known results. We also relate the Cohen-Macaulay property of F to the Cohen-Macaulay property of R and G . |
