Non-Cohen-Macaulay projective monomial curves
We compare several ways of describing how far the homogeneous coordinate ring of a projective monomial curve is from being Cohen-Macaulay. We give a number of examples and then use these ideas to show that the fraction of projective monomial curves of a given degree that are Cohen-Macaulay approaches zero as the degree goes to infinity.
Reid, Les, and Leslie G. Roberts. "Non-Cohen–Macaulay projective monomial curves." Journal of Algebra 291, no. 1 (2005): 171-186.
Journal of Algebra