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Simplicial Toric Varieties Which Are Set-Theoretic Complete Intersections

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Commutative Algebra and its Interactions to Algebraic Geometry

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2210))

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Abstract

We say that a polynomial ideal I is set-theoretically generated by a family of elements f 1, …, f k in I if the radical of I coincides with the radical of the ideal generated by f 1, …, f k. Over an algebraically closed field, the smallest number among all such possible k is the minimal number of equations needed to define the zero set of I. To find this number is a classical problem in both Commutative Algebra and Algebraic Geometry. This problem is even not solved for the defining ideals of toric varieties, whose zeros are given parametrically by monomials. In this lecture notes we study set-theoretically generation of the defining ideals of simplicial toric varieties, which are defined by the property that the exponents of the parametrizing monomials span a simplicial complex. We review and improve most of results on simplicial toric varieties which are set-theoretic complete intersections, previously obtained by the author in collaboration with M. Barile and A. Thoma.

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References

  1. M. Barile, M. Morales, On the equations defining projective monomial curves. Commun. Algebra, 26, 1907–1912 (1998)

    Google Scholar 

  2. M. Barile, M. Morales, A. Thoma, On simplicial toric varieties which are set-theoretic complete intersections. J. Algebra 226(2), 880–892 (2000)

    Google Scholar 

  3. H. Bresinsky, Monomial Gorenstein curves in A 4 as set-theoretic complete intersections. Manuscr. Math. 27, 353–358 (1979)

    Google Scholar 

  4. H.A. Casillas, M. Morales, Sums of toric ideals (2012). arXiv:1211.5386

    Google Scholar 

  5. H. Cohn, Advanced Number Theory, vol. XI (Dover, New York, 1980), 276 p.

    Google Scholar 

  6. L. Coudurier, M. Morales, Classification of courbes toriques dans l’espace projectif, module de Rao et liaison. J. Algebra 211(2), 524–548 (1999)

    Google Scholar 

  7. D. Eisenbud, B. Sturmfels, Binomial ideals. Duke Math. J. 84(1), 1–45 (1996)

    Google Scholar 

  8. K. Eto, Defining ideals of semigroup rings which are Gorenstein. Commun. Algebra 24(12), 3969–3978 (1996)

    Google Scholar 

  9. K. Fischer, W. Morris, J. Shapiro, Affine semigroup rings that are complete intersections. Proc. Am. Math. Soc. 125, 3137–3145 (1997)

    Google Scholar 

  10. R. Hartshorne, Complete intersections in characteristic p > 0. Am. J. Math. 101, 380–383 (1979)

    Google Scholar 

  11. J. Heger, Denkschriften. Kais. Akad. Wissensch. Mathem. Naturw. Klasse 14, II (1858)

    Google Scholar 

  12. E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry (Birkhüuser, Boston, 1985), 238 p.

    MATH  Google Scholar 

  13. T.T. Moh, Set-theoretic complete intersections. Proc. Am. Math. Soc. 94, 217–220 (1985)

    Article  MathSciNet  Google Scholar 

  14. M. Morales, Syzygies of monomial curves and a linear Diophantine problem of Frobenius. Max Planck Institut für Mathematik, Bonn-RFA (1987). Preprint

    Google Scholar 

  15. M. Morales, Equations des Variétés Monomiales en codimension deux. J. Algebra 175, 1082–1095 (1995)

    Article  MathSciNet  Google Scholar 

  16. M. Morales, A. Thoma, Complete intersection lattices ideals. J. Algebra 284(2), 755–770 (2005)

    Article  MathSciNet  Google Scholar 

  17. T.H.N. Nhan, On set-theoretic complete intersection monomial curves in P 3. Lobachevskii J. Math. 34(2), 133–136 (2013)

    Article  MathSciNet  Google Scholar 

  18. T.H.N. Nhan, M. Sahin, Equations defining recursive extensions as set theoretic complete intersections. Tokyo J. Math. 38, 273–282 (2015)

    Article  MathSciNet  Google Scholar 

  19. L. Robbiano, G. Valla, Some curves in P 3 are set-theoretic complete intersections, in Algebraic Geometry - Open Problems. Proceedings of Ravello 1982, ed. by C. Ciliberto, F. Ghione, F. Orecchia. Lecture Notes in Mathematics, vol. 997 (Springer, Berlin, 1983), pp. 391–399

    Google Scholar 

  20. M. Sahin, Producing set-theoretic complete intersection monomial curves in P n. Proc. Am. Math. Soc. 137(4), 1223–1233 (2009)

    Article  MathSciNet  Google Scholar 

  21. M. Sahin, Extensions of toric varieties. Electron. J. Comb. 18(1), 93, 1–10 (2011)

    Google Scholar 

  22. T. Schmitt, W. Vogel, Note on set-theoretic intersections of subvarieties of projective space. Math. Ann. 245, 247–253 (1979)

    Article  MathSciNet  Google Scholar 

  23. A. Schrijver, Theory of Linear and Integer Programming (Wiley, Chichester, 1998), xi, 471 p.

    MATH  Google Scholar 

  24. A. Thoma, On the binomial arithmetical rank. Arch. Math. 74(1), 22–25 (2000)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author thanks Vietnam Institute for Advanced Study in Mathematics (VIASM) and the Institute of Mathematics of Hanoi for their support and hospitality. Sincere thanks are due to Le Tuan Hoa, Ngo Viet Trung, Mesut Sahin and the anonymous referees for many useful suggestions which has improved the presentation of the lecture notes.

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Authors and Affiliations

  1. Université Grenoble Alpes, Institut Fourier, UMR 5582, Saint-Martin D’Hères Cedex, France

    Marcel Morales

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  1. Marcel Morales
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Correspondence to Marcel Morales .

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Editors and Affiliations

  1. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

    Nguyen Tu CUONG

  2. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

    Le Tuan HOA

  3. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

    Ngo Viet TRUNG

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Morales, M. (2018). Simplicial Toric Varieties Which Are Set-Theoretic Complete Intersections. In: Tu CUONG, N., Tuan HOA, L., Viet TRUNG, N. (eds) Commutative Algebra and its Interactions to Algebraic Geometry. Lecture Notes in Mathematics, vol 2210. Springer, Cham. https://doi.org/10.1007/978-3-319-75565-6_4

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