Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The information encoded in initial ideals


Authors: Gunnar Fløystad and Mark L. Green
Journal: Trans. Amer. Math. Soc. 353 (2001), 1427-1453
MSC (2000): Primary 13P10; Secondary 14H50
Published electronically: November 29, 2000
MathSciNet review: 1806734
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We consider homogeneous ideals $I$ and the initial ideal $\text{in}(I)$ for the revlex order. First we give a sequence of invariants computed from $I$ giving better and better ``approximations" to the initial ideal and ending in an equivalent description.

Then we apply this to different settings in algebraic geometry to understand what information is encoded in the generic initial ideal of the ideal of a projective scheme.

We also consider the higher initial ideals as defined in a paper by Fløystad. In particular, we show that giving the generic higher initial ideal of a space curve is equivalent to giving the generic initial ideal of a linked curve.


References [Enhancements On Off] (What's this?)

  • 1. David Bayer and Michael Stillman, A criterion for detecting 𝑚-regularity, Invent. Math. 87 (1987), no. 1, 1–11. MR 862710, 10.1007/BF01389151
  • 2. C. v. Bothmer. Thesis, University of Bayreuth (1995).
  • 3. Michele Cook, The connectedness of space curve invariants, Compositio Math. 111 (1998), no. 2, 221–244. MR 1606169, 10.1023/A:1000316500235
  • 4. David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
  • 5. Gunnar Fløystad, Higher initial ideals of homogeneous ideals, Mem. Amer. Math. Soc. 134 (1998), no. 638, viii+68. MR 1432143, 10.1090/memo/0638
  • 6. Gunnar Fløystad, A property deducible from the generic initial ideal, J. Pure Appl. Algebra 136 (1999), no. 2, 127–140. MR 1674773, 10.1016/S0022-4049(97)00165-5
  • 7. André Galligo, À propos du théorème de-préparation de Weierstrass, Fonctions de plusieurs variables complexes (Sém. François Norguet, octobre 1970–décembre 1973; à la mémoire d’André Martineau), Springer, Berlin, 1974, pp. 543–579. Lecture Notes in Math., Vol. 409 (French). Thèse de 3ème cycle soutenue le 16 mai 1973 à l’Institut de Mathématique et Sciences Physiques de l’Université de Nice. MR 0402102
  • 8. M. Green. Generic initial ideals. Summer school on commutative algebra, Barcelona 16. - 26. July 1996. Accompanying notes, volume II.
  • 9. Laurent Gruson and Christian Peskine, Genre des courbes de l’espace projectif, Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977) Lecture Notes in Math., vol. 687, Springer, Berlin, 1978, pp. 31–59 (French). MR 527229
  • 10. R. Liebling. Classification of space curves using initial ideals. Ph.D. thesis, University of California at Berkeley, (1996).
  • 11. Mireille Martin-Deschamps and Daniel Perrin, Sur la classification des courbes gauches, Astérisque 184-185 (1990), 208 (French). MR 1073438

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13P10, 14H50

Retrieve articles in all journals with MSC (2000): 13P10, 14H50


Additional Information

Gunnar Fløystad
Affiliation: Matematisk Institutt, Johs. Brunsgate 12, 5008 Bergen, Norway
Email: gunnar@mi.uib.no

Mark L. Green
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90024
Email: mlg@math.ucla.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02737-9
Received by editor(s): June 5, 1999
Published electronically: November 29, 2000
Article copyright: © Copyright 2000 American Mathematical Society