Advertisement

Generic Initial Ideals

  • Mark L. Green
Part of the Progress in Mathematics book series (MBC)

Abstract

A very powerful technique in commutative algebra was introduced by Macaulay, who realized that studying the initial terms of elements of an ideal gives one great insight into the algebra and combinatorics of the ideal. The initial ideal depends on the choice of coordinates, but there is an object, the initial ideal in generic coordinates, which is coordinate-independent. Generic initial ideals appeared in the work of Grauert and Hironaka.

Keywords

Complete Intersection Hilbert Function Monomial Ideal Exterior Algebra Homogeneous Ideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    M. Amasaki, “Preparatory structure theorem for ideals defining space curves,” Publ. RIMS 19 (1983), 493–518.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [A-C-G-H]
    E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of Algebraic Curves, V. 1, Springer-Verlag, New York, 1985.zbMATHGoogle Scholar
  3. [A-H-H]
    A. Aramova, J. Herzog and T. Hibi, “Gotzmann theorems for exterior algebras and combinatorics,” preprint.Google Scholar
  4. [B-C-G3]
    R. Bryant, S.S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths, Exterior Differential Systems, Springer-Verlag, Boston, 1991.zbMATHCrossRefGoogle Scholar
  5. [B-H]
    W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1993.zbMATHGoogle Scholar
  6. [B-S]
    D. Bayer and M. Stillman, “A criterion for detecting m-regularity,” Invent. Math. 87 (1987), 1–11.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [C]
    M. Cook, “The connectedness of space curve invariants,” preprint.Google Scholar
  8. [D]
    E. Davis, “O-dimensional subschemes of P2:lectures on Castelnuovo’s function,” Queen’s Papers in pure and appl. math. 76 (1986).Google Scholar
  9. [E]
    D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York 1995.zbMATHGoogle Scholar
  10. [E-G-H]
    D. Eisenbud, M. Green, and J. Harris, “Some conjectures extending Castelnuovo theory,” in Astérisque 218 (1993), 187–202.MathSciNetGoogle Scholar
  11. [E-K]
    S. Eliahou and M. Kervaire, “Minimal resolutions of some monomial ideals,” J. Alg. 129 (1990), 1–25.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [E-P]
    Ph. Ellia and C. Peskine, “Groupes de points de P2: caractère et position uniforme,” 111–116 in Algebraic geometry (L’Aquila, 1988), Springer LNM 1417, Berlin, 1990.Google Scholar
  13. [F]
    G. Floystad, “Higher initial images of homogeneous ideals,” preprint.Google Scholar
  14. [G1]
    M. Green, “Koszul cohomology and geometry,” 177–200 in Lectures on Riemann Surfaces (M. Cornalba, X. Gomez-Mont, A. Verjovsky eds.), World Scientific, Singapore, 1989.Google Scholar
  15. [G2]
    M. Green, “Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann,” 76–88 in Algebraic Curves and Projective Geometry (E. Ballico and C. Ciliberto, eds.), Springer LNM 1389, Berlin 1989.Google Scholar
  16. [G3]
    M. Green, “Infinitesimal methods in Hodge theory,” 1–92 in Algebraic Cycles and Hodge Theory (A. Albano and F. Bardelli, eds.), Springer LNM 1594, Berlin, 1994.Google Scholar
  17. [G4]
    M. Green, “The Eisenbud-Koh-Stillman conjecture on linear syzygies,” preprint.Google Scholar
  18. [G-H]
    P. Griffiths, and J. Harris, Principles of Algebraic Geometry, Wiley Interscience, New York, 1978.zbMATHGoogle Scholar
  19. [G-H2]
    P. Griffiths and J. Harris, “Algebraic geometry and local differential geometry,” Ann. Sci. École Norm. Sup. 12 (1979), 355–452.MathSciNetzbMATHGoogle Scholar
  20. [G-L-P]
    L. Gruson, R. Lazarsfeld, and C. Peskine, “On a theorem of Castelnuovo, and the equations defining space curves,” Inv. Math. 72 (1983), 491–506.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [G-P]
    L. Gruson and C. Peskine, “Genre des courbes de l’espace projectif,” 31–59 in Algebraic Geometry: Proc. Symposium University of Tromso, Tromso, 1977, Springer LNM 687, Berlin, 1978.Google Scholar
  22. [Go]
    G. Gotzmann, “Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes,” Math. Z. 158 (1978), 61–70.MathSciNetCrossRefGoogle Scholar
  23. [H]
    R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.zbMATHCrossRefGoogle Scholar
  24. [H-U]
    C. Huneke and B. Ulrich, “General hyperplane sections of algebraic varieties,” J. Alg. Geom. 2 (1993), 487–505.MathSciNetzbMATHGoogle Scholar
  25. [L]
    J.M. Landsberg, “On the second fundamental forms of projective varieties,” Inv. Math. 117 (1994), 303–315.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [L2]
    J.M. Landsberg, “Differential-geometric characterizations of complete intersections,” preprint.Google Scholar
  27. [Li]
    R. Liebling, “Classification of space curves using initial ideals,” Thesis, University of California at Berkeley, 1996.Google Scholar
  28. [M]
    H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.zbMATHGoogle Scholar
  29. [Ri]
    C. Rippel, “Generic initial ideal theory for coordinate rings of flag varieties,” Thesis (UCLA).Google Scholar
  30. [Rol]
    L. Robbiano, “Term orderings in the polynomial ring,” Proc. of Eurocal 1985, LNCS 203 II, Springer-Verlag, Berlin, 1985.Google Scholar
  31. [Ro2]
    L. Robbiano, “On the theory of graded structures,” J. Symb. comp 2 (1986), 139–170.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [R]
    J.-E. Roos, “A description of the homological behavior of families of quadratic forms in four variables,” in Iarrobino, Martsinkovsky, and Weyman, Syzygies and Geometry (1995), 86–95.Google Scholar
  33. [St]
    J.-P. Serre, “Faisceaux algébriques cohérents,” Ann. of Math. 61 (1955), 197–278.Google Scholar
  34. [St]
    R. Stanley, Combinatorics and Commutative Algebra, 2nd edition, Birkhäuser, Boston, 1996.zbMATHGoogle Scholar
  35. [Str]
    R. Strano, “A characterization of complete intersection curves in P3,” Proc. A.M.S. 104 (1988), no. 3, 711–715.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 1998

Authors and Affiliations

  • Mark L. Green
    • 1
  1. 1.University of CaliforniaLos AngelesUSA

Personalised recommendations