Maximal and Cohen-Macaulay projective monomial curves
Let \( {\mathcal{S}}=\{a_1,\dots , a_k \}\) be a sequence of integers with \(0<a_1<\dots < a_k=d\) and \(\text{ gcd}( a_1,\dots ,a_k)=1\). Let \(R_{\mathcal{S} }= K[s^d, s^{d-a_1}t^{a_1}, s^{d-a_2}t^{a_2}, \dots ,t^d]\) be the homogeneous coordinate ring of a projective monomial curve over a field \(K\). Several authors [see e.g. \textit{S. Goto, N. Suzuki} and \textit{K. Watanabe}, Jap. J. Math., New. Ser. 2, 1--12 (1976; Zbl 0361.20066)] have given necessary and sufficient conditions for \(R_{\mathcal{S} }\) to be Cohen-Macaulay. In the present paper the authors continue the investigation of the Cohen-Macaulay property of the ring \(R_{\mathcal{S}}\) begun in [J. Algebra 291, No. 1, 171--186 (2005; Zbl 1085.14042)]. With this aim, the authors introduce the so-called maximal curves, observing that every Cohen-Macaulay projective monomial curve is maximal, but not conversely and that the test for maximality is simpler than for Cohen-Macaulayness. In particular they study closely the asymptotic properties of maximal and Cohen-Macaulay curves and show that the number of all curves of degree \(d\) that are Cohen-Macaulay grows exponentially in \(d\), but not as fast as the total number of curves and that maximal curves of degree \(d\) with sufficiently large embedding dimension relative to \(d\) are Cohen-Macaulay.
scientific article; zbMATH DE number 5118826
scientific article