TAILIEUCHUNG - On symmetric monomial curves in P3
In this paper, we give an elementary proof of the fact that symmetric arithmetically Cohen-Macaulay monomial curves are set-theoretic complete intersections. The proof is constructive and provides the equations of the surfaces cutting out the monomial curve. | Turk J Math 33 (2009) , 107 – 110. ¨ ITAK ˙ c TUB doi: On Symmetric Monomial curves in P3 Mesut S ¸ ahin Abstract In this paper, we give an elementary proof of the fact that symmetric arithmetically Cohen-Macaulay monomial curves are set-theoretic complete intersections. The proof is constructive and provides the equations of the surfaces cutting out the monomial curve. Key Words: Set-theoretic complete intersections, monomial curves 1. Introduction Let K be an algebraically closed field and R be the polynomial ring K[x0 , . . . , xn] . To any irreducible curve C in Pn , one can associate a prime ideal I(C) ⊂ R to be the set of all polynomials vanishing on C . The arithmetical rank of C , denoted by μ(C), is the least positive integer r for which I(C) = rad(f1 , . . . , fr ), for some polynomials f1 , . . . , fr or equivalently C = H1 · · · Hr , where H1 , . . . , Hr are the hypersurfaces defined by f1 = 0, · · · , fr = 0 , respectively. We denote by μ(I(C)) the minimal number r for which I(C) = (f1 , . . . , fr ), for some polynomials f1 , . . . , fr ∈ R . These invariants are known to be bounded below by the codimension of the curve (or height of its ideal). So, one has the following relation: n − 1 ≤ μ(C) ≤ μ(I(C)) Although μ(I(C)) has no upper bound (see [1], for an example), an upper bound for μ(C) is provided to be n in [7] via commutative algebraic methods. Later in [2, 22] the equations of these n hypersurfaces that cut out the curve C were given explicitly by using elementary algebraic methods. The curve C is called a complete intersection if μ(I(C)) = n − 1 . It is called an almost complete intersection, if instead, one has μ(I(C)) = n. When the arithmetical rank of C takes its lower bound, that is μ(C) = n − 1 , the curve C is called a set-theoretic complete intersection, . for short. It is clear that complete intersections are set-theoretic complete intersection. The corresponding question for almost .
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