Beiträge zur Algebra und Geometrie <BR> Contributions to Algebra and GeometryVol. 40, No. 2, pp. 437-458 (1999)

# On the Cohen-Macaulayness of the Coordinate Ring of Certain Projective Monomial Curves

## S. Molinelli, D. P. Patil, G. Tamone

Dipartimento di Matematica, Universita di Genova, via Dodecaneso 35, I-16146 Genova, Italy, e-mail: tamone@dima.unige.it, molinell@.dima.unige.it

Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India, e-mail: patil@math.iisc.ernet.in

Abstract: \font\msbm=msbm10 \def\PP{\hbox{\msbm P}} Let $K$ be a field and let $n_1,\ldots,n_e$ be a sequence of positive integers with $\gcd(n_1,\ldots,n_e)=1$ and $n_1<n_2<\cdots<n_e$. Let $A'$ be the coordinate ring of the projective monomial curve in the projective $e$-space $\PP_K^e$ 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 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 assumption 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.

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