Non-Cohen-Macaulay projective monomial curves. (English) Zbl 1085.14042
The authors consider the homogeneous coordinate ring
\[
R = R_{\mathcal S} = K[s^d, s^{d-a_1} t^{a_1}, \dots, s^{d-a_{k-1}} t^{a_{k-1}}, t^d ]
\]
of a projective monomial curve over a field \(K\) defined by a sequence \(\mathcal S = \{a_1,\dots , a_k\}\) of positive integers with \(0 < a_1 < \dots < a_k = d\).
The main result of this paper is to compare several ways of describing how far the ring \(R\) is from being Cohen-Macaulay. Several authors (for example S. Goto, N. Suzuki, K. Watanabe [Jap. J. Math., New. Ser. 2, 1–12 (1976; Zbl 0361.20066)], Ngô Viêt Trung and Lê Tuân Hoa [Trans. Am. Math. Soc. 298, 145–167 (1986; Zbl 0631.13020)]) have given necessary and sufficient conditions for \(R\) to be Cohen-Macaulay in terms of unstable elements or the cardinalities of \(\tilde S \setminus S\) or \(S'\setminus S\), where \(S\) is the semigroup generated by \(\{(d, 0), (d-a_1, a_1),\dots, (d-a_{k-1}, a_{k-1}), (0, d)\}\), \(G := \{(x, y)\in\mathbb Z^2 \mid x +y \equiv 0 \mod d\}\), \(\tilde S := \{\sigma\in G \mid \sigma +(0, d)\in S\text{ and }\sigma +(d, 0)\in S\}\) and \(S' := \{\sigma\in G\mid \sigma+ a(0, d)\in S\text{ and }\sigma + b(d, 0)\in S\text{ for some }a, b \geq 0\}\).
In section 2, the authors give several ways of describing how far \(R\) is from being Cohen-Macaulay in terms of the spanning set \(\mathcal B\) of \(S\) over \(T\), where \(T\) is the subsemigroup of \(S\) generated by \((d, 0)\) and \((0, d)\). The spanning set of \(S\) over \(T\) is defined in the Ph. D. thesis of Ping Li [Queen’s University, 2005] and is the set \(\mathcal B := \{\sigma\in S\mid\sigma\text{ cannot be written in the form }\sigma'+\tau\text{ with }\tau\in T,\sigma' \in S\}\).
In section 3 the theorem 2.1 is illustrated by an easy example. In section 4 the results of section 2 are applied to complicated examples of the reviewer’s joint article with L.G. Roberts [J. Pure Appl. Algebra 183, No.1–3, 275–292 (2003; Zbl 1078.13509)], gaining new insight into what is special about these examples.
In section 5 it is proved that as \(d\) goes to infinity the fraction of the rings \(R_{\mathcal S}\) that are Cohen-Macaulay goes to 0 (as \(\mathcal S\) ranges over all sequences \(\{a_1,\dots, a_k\}\) with \(0 < a_1 <\dots < a_k = d\). In section 6 some numerical results are presented which illustrate the results.
The main result of this paper is to compare several ways of describing how far the ring \(R\) is from being Cohen-Macaulay. Several authors (for example S. Goto, N. Suzuki, K. Watanabe [Jap. J. Math., New. Ser. 2, 1–12 (1976; Zbl 0361.20066)], Ngô Viêt Trung and Lê Tuân Hoa [Trans. Am. Math. Soc. 298, 145–167 (1986; Zbl 0631.13020)]) have given necessary and sufficient conditions for \(R\) to be Cohen-Macaulay in terms of unstable elements or the cardinalities of \(\tilde S \setminus S\) or \(S'\setminus S\), where \(S\) is the semigroup generated by \(\{(d, 0), (d-a_1, a_1),\dots, (d-a_{k-1}, a_{k-1}), (0, d)\}\), \(G := \{(x, y)\in\mathbb Z^2 \mid x +y \equiv 0 \mod d\}\), \(\tilde S := \{\sigma\in G \mid \sigma +(0, d)\in S\text{ and }\sigma +(d, 0)\in S\}\) and \(S' := \{\sigma\in G\mid \sigma+ a(0, d)\in S\text{ and }\sigma + b(d, 0)\in S\text{ for some }a, b \geq 0\}\).
In section 2, the authors give several ways of describing how far \(R\) is from being Cohen-Macaulay in terms of the spanning set \(\mathcal B\) of \(S\) over \(T\), where \(T\) is the subsemigroup of \(S\) generated by \((d, 0)\) and \((0, d)\). The spanning set of \(S\) over \(T\) is defined in the Ph. D. thesis of Ping Li [Queen’s University, 2005] and is the set \(\mathcal B := \{\sigma\in S\mid\sigma\text{ cannot be written in the form }\sigma'+\tau\text{ with }\tau\in T,\sigma' \in S\}\).
In section 3 the theorem 2.1 is illustrated by an easy example. In section 4 the results of section 2 are applied to complicated examples of the reviewer’s joint article with L.G. Roberts [J. Pure Appl. Algebra 183, No.1–3, 275–292 (2003; Zbl 1078.13509)], gaining new insight into what is special about these examples.
In section 5 it is proved that as \(d\) goes to infinity the fraction of the rings \(R_{\mathcal S}\) that are Cohen-Macaulay goes to 0 (as \(\mathcal S\) ranges over all sequences \(\{a_1,\dots, a_k\}\) with \(0 < a_1 <\dots < a_k = d\). In section 6 some numerical results are presented which illustrate the results.
Reviewer: D. P. Patil (Bangalore)
MSC:
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
References:
| [1] | Bruns, W.; Gubeladze, J.; Trung, N., Problems and algorithms for affine semigroups, Semigroup Forum, 64, 180-212 (2002) · Zbl 1018.20048 |
| [2] | Bruns, W.; Herzog, J., Cohen-Macaulay Rings (1993), Cambridge University Press · Zbl 0788.13005 |
| [3] | Derksen, H.; Kemper, G., Computational Invariant Theory, Encyclopedia Math. Sci. (2002), Springer · Zbl 1011.13003 |
| [4] | Goto, S.; Suzuku, N.; Watanabe, K., On affine semigroup rings, Japan J. Math., 2, 1-12 (1976) · Zbl 0361.20066 |
| [5] | P. Li, PhD thesis, Queen’s University, 2005; P. Li, PhD thesis, Queen’s University, 2005 |
| [6] | Patil, D. P.; Roberts, L. G., Hilbert functions of monomial curves, J. Pure Appl. Algebra, 183, 275-292 (2002) · Zbl 1078.13509 |
| [7] | Roberts, L. G., Certain projective curves with unusual Hilbert function, J. Pure Appl. Algebra, 104, 303-311 (1995) · Zbl 0842.13011 |
| [8] | Trung, L.; Hoa, L., Affine semigroups and Cohen-Macaulay rings generated by monomials, Trans. Amer. Math. Soc., 298, 145-167 (1986) · Zbl 0631.13020 |
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