Set-theoretic complete intersections on binomials

Authors:
Margherita Barile, Marcel Morales and Apostolos Thoma

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1893-1903

MSC (2000):
Primary 14M25, 13C40, 14M10

DOI:
https://doi.org/10.1090/S0002-9939-01-06289-X

Published electronically:
December 20, 2001

MathSciNet review:
1896020

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an affine toric variety of codimension 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, is a set-theoretic complete intersection on binomials if and only if is a complete intersection. Moreover, if are binomials such that , then . While in the positive characteristic case, is a set-theoretic complete intersection on binomials if and only if is completely -glued.

These results improve and complete all known results on these topics.

**1.**M. Barile, M. Morales, A. Thoma,*On Simplicial Toric varieties which are Set-Theoretic Complete Intersections*, Journal of Algebra 226 (2000) 880-892. CMP**2000:11****2.**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.**3.**E. Becker, R. Grobe, M. Niermann,*Radicals of binomial ideals*, J. Pure and Applied Algebra 117 & 118 (1997) 41-79. MR**98k:13029****4.**Ch. Delorme,*Sous monoides d' intersection complete de*, Ann. scient. Ec. Norm. Sup., 4 serie, 9 (1976) 145-154. MR**53:10821****5.**D. Eisenbud and B. Sturmfels,*Binomial ideals*, Duke Math. J. 84 (1996) 1-45. MR**97d:13031****6.**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.**7.**K. Fischer, J. Shapiro,*Mixed matrices and binomials ideals*, J. Pure and Applied Algebra 113 (1996) 39-54. MR**97h:13008****8.**K. Fischer, W. Morris, J. Shapiro,*Affine semigroup rings that are complete intersections*, Proc. Amer. Math. Soc. 125 (1997) 3137-3145. MR**97m:13026****9.**R. Hartshorne,*Complete Intersections in characteristic*, Amer. J. Math. 101 (1979) 380-383. MR**80d:14028****10.**T. T. Moh,*Set-theoretic complete intersections*, Proc. Amer. Math. Soc. 94 (1985) 217-220. MR**86e:14026****11.**J. C. Rosales,*On Presentations of subsemigroups of*, Semigroup Forum 55 (1997) 152-159. MR**98h:20104****12.**J.C. Rosales and P. A. Garcia-Sanchez,*On complete intersection affine semigroups*, Commun. Algebra 23(14) (1995) 5395-5412. MR**96m:14068****13.**G. Scheja, O. Scheja, U. Storch,*On regular sequences of binomials*, Manuscripta Math. 98 (1999) 115-132. MR**99k:13017****14.**B. Sturmfels, Gröbner Bases and Convex Polytopes. University Lecture Series, No. 8 American Mathematical Society Providence, R.I. 1995. MR**97b:13034****15.**A. Thoma,*On the binomial arithmetical rank*, Arch. Math. 74 (2000) 22-25. MR**2001a:14023**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
14M25,
13C40,
14M10

Retrieve articles in all journals with MSC (2000): 14M25, 13C40, 14M10

Additional Information

**Margherita Barile**

Affiliation:
Dipartimento di Matematica, Università degli Studi di Bari, Via Orabona 4, 70125 Bari, Italy

Email:
barile@dm.uniba.it

**Marcel Morales**

Affiliation:
Université de Grenoble I, Institut Fourier, UMR 5582, B.P.74, 38402 Saint-Martin D’Hères Cedex, and IUFM de Lyon, 5 rue Anselme, 69317 Lyon Cedex, France

Email:
Marcel.Morales@ujf-grenoble.fr

**Apostolos Thoma**

Affiliation:
Department of Mathematics, Purdue Univerity, West Lafayette, Indiana 47907-1395

Address at time of publication:
Department of Mathematics, University of Ioannina, Ioannina 45110, Greece

Email:
athoma@cc.uoi.gr

DOI:
https://doi.org/10.1090/S0002-9939-01-06289-X

Keywords:
Affine semigroups,
binomial ideals,
complete intersections,
set-theoretic complete intersections,
toric varieties

Received by editor(s):
October 17, 2000

Received by editor(s) in revised form:
January 16, 2001

Published electronically:
December 20, 2001

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 2001
American Mathematical Society