Hilbert functions of monomial curves

Non-Cohen–Macaulay projective monomial curves

“…, m p , m p+1 } where δ, m = m 0 are relatively prime, and m i = m + iδ, 0 i p + 1. This case was studied in [6], where the authors found explicitly the degrees of all elements of Γ , and hence the unstable elements, and Hilbert function for curves arising from S . (Here we use δ in place of d, which was used in [6] for this quantity, to avoid conflict with our use of d for degree.…”

“…This case was studied in [6], where the authors found explicitly the degrees of all elements of Γ , and hence the unstable elements, and Hilbert function for curves arising from S . (Here we use δ in place of d, which was used in [6] for this quantity, to avoid conflict with our use of d for degree. Of course d = m p+1 and k = p + 3, but we will use m and p in this section to facilitate use of results from [6].)…”

“…(Here we use δ in place of d, which was used in [6] for this quantity, to avoid conflict with our use of d for degree. Of course d = m p+1 and k = p + 3, but we will use m and p in this section to facilitate use of results from [6].) As in previous sections of this paper S S (or simply S) will denote the semigroup in N 2 constructed from S in Section 1, Γ will denote the projection of S onto the second coordinate andΓ be the projection onto the first coordinate.…”

“…In Section 3 we illustrate our theory with an easy example. Then in Section 4 we apply our results to the more complicated examples of [6], gaining new insight into what is special about these examples. Then in Section 5 we prove that as d goes to infinity the fraction of the rings R S that are Cohen-Macaulay goes to 0 (as S ranges over all sequences {a 1 , .…”

“…Since {s d , t d } is a homogeneous system of parameters of R S , R S is Cohen-Macaulay if and only if t d is a non-zero-divisor in gr(S ). Integers x ∈ Γ were defined in [6] to be unstable if t x is killed by some power of t d in gr(S ), equivalently there exists a positive integer i such that ord S (x + id) < ord S (x) + i. Let T be the subsemigroup of S generated by (d, 0) and (0, d).…”