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The pinched Veronese is Koszul

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Abstract

In this paper we prove that the coordinate ring of the pinched Veronese, k[X 3,X 2 Y,XY 2,Y 3,X 2 Z,Y 2 Z,XZ 2,YZ 2,Z 3], is Koszul. The result is obtained by combining the use of a flat deformation induced by a distinguished weight together with a generalization of the notion of Koszul filtrations.

Keywords

Koszul algebras Pinched Veronese Koszul filtration 
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References

  1. 1.
    Anders, N.J.: CaTS, a software package for computing state polytopes of toric ideals, available from http://www.soopadoopa.dk/anders/cats/cats.html
  2. 2.
    Blum, S.: Subalgebras of bigraded Koszul algebras. J. Algebra 242(2), 795–809 (2001) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bruns, W., Herzog, J., Vetter, U.: Syzygies and walks. In: Simis, A., Trung, N.V., Valla, G. (eds.) ICTP Proceedings ‘Commutative Algebra’, pp. 36–57. World Scientific, Singapore (1994) Google Scholar
  4. 4.
    Bruns, W., Gubeladze, J., Trung, N.V.: Normal polytopes, triangulations, and Koszul algebras. J. Reine Angew. Math. 485, 123–160 (1997) MATHMathSciNetGoogle Scholar
  5. 5.
    Conca, A., Herzog, J.: Castelnuovo-Mumford regularity of products of ideals. Preprint Google Scholar
  6. 6.
    Conca, A., Herzog, J., Trung, N.V., Valla, G.: Diagonal subalgebras of a bigraded algebras and embeddings of blow ups of projective spaces. Amer. J. Math. 119, 859–901 (1997) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Conca, A., Trung, N.V., Valla, G.: Koszul property for points in projective spaces. Math. Scand. 89, 201–216 (2001) MATHMathSciNetGoogle Scholar
  8. 8.
    Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, New York (1995) MATHGoogle Scholar
  9. 9.
    Eisenbud, D., Reeves, A., Totaro, B.: Initial ideals, Veronese subrings, and rates of algebras. Adv. Math. 109, 168–187 (1994) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fröberg, R.: Koszul algebras. In: Advances in Commutative Ring Theory (Fez, 1997). Lecture Notes in Pure and Appl. Math., vol. 205, pp. 337–350. Dekker, New York (1999) Google Scholar
  11. 11.
    Grayson, D., Stillman, M.: Macaulay 2: a software system for algebraic geometry and commutative algebra available over the web at http://www.math.uiuc.edu/Macaualy2
  12. 12.
    Herzog, J., Hibi, T., Restuccia, G.: Strongly Koszul algebras. Math. Scand. 86, 161–178 (2000) MATHMathSciNetGoogle Scholar
  13. 13.
    Matsumura, H.: Commutative Ring Theory, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989). Translated from the Japanese by M. Reid MATHGoogle Scholar
  14. 14.
    Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence (1996), xii+162 pp. MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Mathematics DepartmentPurdue UniversityWest LafayetteUSA

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