On symmetric monomial curves in $Bbb{P}^3$

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.

PDF

___

[1] Arslan, F.: Cohen-Macaulayness of tangent cones, Proc. Amer. Math. Soc. 128, 2243-2251 (2000).
[2] Barile, M. and Morales, M.: On the equations defining projective monomial curves, Comm. Algebra 26, 1907-1912 (1998).
[3] Barile, M., Morales, M. and Thoma, A.: On simplicial toric varieties which are set-theoretic complete intersections, Journal of Algebra 226, 880-892 (2000).
[4] Bresinsky, H.: Monomial space curves in A3 as set-theoretic complete intersection, Proc. Amer. Math. Soc. 75, 23-24 (1979).
[5] Bresinsky, H.: Monomial Gorenstein curves in A4 as set-theoretic complete intersectionv Manuscripta Math. 27,353-358(1979).
[6] Bresinsky, H., Schenzel, P. and Vogel, W.: On liaison, arithmetical Buchsbaum curves and monomial curves in P3, Journal of Algebra 86, 283-301 (1984).
[7] Eisenbud, D. and Evans, E.G.: Every algebraic set in n-space is the intersection of n hypersurfaces, Inventiones Math. 19, 107-112 (1973).
[8] Eto, K.: Almost complete intersection monomial curves in A4 , Comm. Algebra 22, 5325-5342 (1994).
[9] Eto, K.: Almost complete intersection monomial curves in A5 , Gakujutsu Kenkyu (Academic Studies) Math. Waseda Univ. 43, 35-47 (1995).
[10] Eto, K.: Almost Complete Intersection Lattice Ideals, Report of researches 35(2), 237-248 (2005/9).
[11] Eto, K.: Set-theoretic complete intersection lattice ideals in monoid rings, Journal of Algebra 299, 689-706 (2006).
[12] Hartshorne, R.: Complete intersections in characteristic p > 0, Amer. J. Math. 101, 380-383 (1979).
[13] Katsabekis, A.: Projection of cones and the arithmetical rank of toric varieties, J. Pure Appl. Algebra 199, 133-147 (2005).
[14] Moh, T.T.: Set-theoretic complete intersections, Proc. Amer. Math. Soc. 94, 217-220 (1985).
[15] Morales, M. and Thoma, A.: Complete intersection lattice ideals, Journal of Algebra 284, 755-770 (2005).
[16] Robbiano, L. and Valla, G.: On set-theoretic complete intersections in the projective space, Rend. Sem. Mat. Fis. Milano LIII, 333-346 (1983).
[17] Robbiano, L. and Valla, G.: Some curves in P3 are set-theoretic complete intersections, in: Algebraic Geometry-Open problems, Proceedings Ravello 1982, Lecture Notes in Mathematics, Vol 997 (Springer, New York), 391-346 ar (1983).
[18] Thoma, A.: On the arithmetically Cohen-Macaulay property for monomial curves, Comm. Algebra 22(3), 805-821(1994).
[19] Thoma, A.: On the set-theoretic complete intersection problem for monomial curves in An and Pn, J. Pure Appl. Algebra 104, 333-344 (1995).
[20] Thoma, A.: Affme semigroup rings and monomial varieties, Comm. Algebra 24(7), 2463-2471 (1996).P
[21] Thoma, A.: Construction of set-theoretic complete intersections via semigroup gluing, Contributions to Algebra and Geometry 41(1), 195-198 (2000).
[22] Thoma, A.: On the binomial arithmetical rank, Arch. Math. 74, 22-25 (2000).