Non-Cohen–Macaulay projective monomial curves

Abstract

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 CohenMacaulay approaches zero as the degree goes to infinity.

Maximal and Cohen–Macaulay projective monomial curves

“…The equivalence of the two statements in Definition 1.2(b) is easy given the previous discussion, and was stated in [6,Lemma 2.6]. Note that S is a semigroup.…”

“…Since only the largest element S 1 of F can be Cohen-Macaulay this suggests that as d goes to infinity the fraction of the curves that are Cohen-Macaulay should approach 0. This indeed was proved to be the case in [6], but by a different approach, without using the sets F .…”

“…, a k ) = 1. We saw in [6] that lim d→∞ c d /2 d−1 = 1, so it will suffice to show that lim d→∞ m d /2 d−1 = 1. Let S = {a 1 , .…”

“…Also asymptotic questions about maximality seem easier. For example, we proved in [6] that the fraction of all curves of degree d that are Cohen-Macaulay approaches 0 as d → ∞ by showing that the fraction that are maximal approaches 0 (although without using the terminology maximal). In this paper we study more closely the asymptotic properties of maximal and Cohen-Macaulay curves.…”

“…Now we consider the examples of [4]. As in [6], we will replace the d's in [4] by δ to avoid conflict with our use of d to represent degree. δ and m, and thatŜ = {δ, 2δ, .…”