Set-theoretic complete intersections on binomials

Barile, Margherita; Morales, Marcel; Thoma, Apostolos

doi:10.1090/S0002-9939-01-06289-X

Set-theoretic complete intersections on binomials
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by Margherita Barile, Marcel Morales and Apostolos Thoma PDF

Proc. Amer. Math. Soc. 130 (2002), 1893-1903 Request permission

Abstract:

Let $V$ be an affine toric variety of codimension $r$ over a field of any characteristic. We completely characterize the affine toric varieties that are set-theoretic complete intersections on binomials. In particular we prove that in the characteristic zero case, $V$ is a set-theoretic complete intersection on binomials if and only if $V$ is a complete intersection. Moreover, if $F_1,\dots ,F_r$ are binomials such that $I(V) = rad(F_1,\dots , F_r)$, then $I(V) = (F_1,\dots ,F_r)$. While in the positive characteristic $p$ case, $V$ is a set-theoretic complete intersection on binomials if and only if $V$ is completely $p$-glued. These results improve and complete all known results on these topics.

References

M. Barile, M. Morales, A. Thoma, On Simplicial Toric varieties which are Set-Theoretic Complete Intersections, Journal of Algebra 226 (2000) 880-892.
M. Barile, M. Morales, A. Thoma, Set-Theoretic Complete Intersections on binomials, the simplicial toric case, Pesquimat, Universidad Nacional de San Marcos, Lima, Peru, November 2000.
Eberhard Becker, Rudolf Grobe, and Michael Niermann, Radicals of binomial ideals, J. Pure Appl. Algebra 117/118 (1997), 41–79. Algorithms for algebra (Eindhoven, 1996). MR 1457832, DOI 10.1016/S0022-4049(97)00004-2
Charles Delorme, Sous-monoïdes d’intersection complète de $N.$, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 1, 145–154. MR 407038
David Eisenbud and Bernd Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), no. 1, 1–45. MR 1394747, DOI 10.1215/S0012-7094-96-08401-X
S. Eliahou and R. Villarreal, On systems of binomials in the ideal of a toric variety, LMPA 96, Laboratoire de Mathématiques Pures et Appliquées, Joseph Liouville, Calais, France, 1999.
Klaus G. Fischer and Jay Shapiro, Mixed matrices and binomial ideals, J. Pure Appl. Algebra 113 (1996), no. 1, 39–54. MR 1411645, DOI 10.1016/0022-4049(95)00144-1
Klaus G. Fischer, Walter Morris, and Jay Shapiro, Affine semigroup rings that are complete intersections, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3137–3145. MR 1401741, DOI 10.1090/S0002-9939-97-03920-8
Robin Hartshorne, Complete intersections in characteristic $p>0$, Amer. J. Math. 101 (1979), no. 2, 380–383. MR 527998, DOI 10.2307/2373984
T. T. Moh, Set-theoretic complete intersections, Proc. Amer. Math. Soc. 94 (1985), no. 2, 217–220. MR 784166, DOI 10.1090/S0002-9939-1985-0784166-1
J. C. Rosales, On presentations of subsemigroups of $\textbf {N}^n$, Semigroup Forum 55 (1997), no. 2, 152–159. MR 1457760, DOI 10.1007/PL00005916
J. C. Rosales and Pedro A. García-Sánchez, On complete intersection affine semigroups, Comm. Algebra 23 (1995), no. 14, 5395–5412. MR 1363611, DOI 10.1080/00927879508825540
Günter Scheja, Ortwin Scheja, and Uwe Storch, On regular sequences of binomials, Manuscripta Math. 98 (1999), no. 1, 115–132. MR 1669583, DOI 10.1007/s002290050129
Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949, DOI 10.1090/ulect/008
Apostolos Thoma, On the binomial arithmetical rank, Arch. Math. (Basel) 74 (2000), no. 1, 22–25. MR 1728358, DOI 10.1007/PL00000405