zbMATH — the first resource for mathematics

On the Cohen-Macaulayness of the coordinate ring of certain projective monomial curves. (English) Zbl 0951.13007
Summary: Let \(K\) be a field and let \(\{n_1,\dots,n_e\}\subset \mathbb{Z}>0\) such that \(\text{gcd}(n_1,\ldots,n_e)=1\) and \(n_1<n_2<\cdots<n_e\). Let \(A'\) be the coordinate ring of the projective monomial curve in the projective \(e\)-space \(\mathbb{P}_K^e\) defined parametrically by \(Z_0=X^{n_e},\ldots,Z_i=X^{n_e-n_i}Y^{n_i},\ldots,Z_{e}=Y^{n_e}\) where \(n_0:=0\). In this paper under some assumptions, we discuss when exactly the graded ring \(A'\) is Cohen-Macaulay and we give a numerical criterion for this in terms of the standard basis of the semigroup generated by \(n_1,\ldots,n_e\) in the case when some \(e-1\) terms of \(n_1,\ldots,n_e\) form an arithmetic sequence. Our special assumptions are satisfied in the case \(e=3\), in particular, for the class of monomial projective space curves, we get a criterion for arithmetically Cohen-Macaulayness.

13C14 Cohen-Macaulay modules
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14H50 Plane and space curves
Full Text: EMIS EuDML