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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.

##### MSC:
 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
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