Remote Access Transactions of the American Mathematical Society
Green Open Access

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

Abstract | References | Similar Articles | Additional Information


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. D. Bayer, M. Stillman. A criterion for detecting m-regularity. Inventiones Matematicae 87 (1987), 1-11. MR 87k:13019
  • 2. C. v. Bothmer. Thesis, University of Bayreuth (1995).
  • 3. M. Cook. The connectedness of space curve invariants. Compositio Mathematica 111 no.2 (1998). MR 99a:14040
  • 4. D. Eisenbud. Commutative algebra with a view towards algebraic geometry. Springer-Verlag, 1995. MR 97a:13001
  • 5. G. Fløystad. Higher initial ideals of homogeneous ideals. Memoirs of the American Mathematical Society 134 no. 638 (1998). MR 98m:13021
  • 6. G. Fløystad. A property deducible from the generic initial ideal. Journal of Pure and Applied Algebra 136 (1999), 127-140. MR 2000c:13039
  • 7. A. Galligo. A propos du theoreme de preparation de Weierstrass. Fonctions de Plusieurs Variables Complexes (Sém. François Norquet, 1970-1972), Lecture Notes in Mathematics, vol. 409, Springer-Verlag, 1973, pp. 543-579. MR 53:5924
  • 8. M. Green. Generic initial ideals. Summer school on commutative algebra, Barcelona 16. - 26. July 1996. Accompanying notes, volume II.
  • 9. L. Gruson, C. Peskine. Genre des courbes de l'espace projectif, Algebraic Geometry: Proc. Symposium University of Tromsø, 1977, Lecture Notes in Mathematics, vol. 687, Springer-Verlag, 1978, pp. 31-59. MR 81e:14019
  • 10. R. Liebling. Classification of space curves using initial ideals. Ph.D. thesis, University of California at Berkeley, (1996).
  • 11. M. Martin-Deschamps, D. Perrin. Sur la classification des courbes gauches. Astérisque 184-185 (1990). MR 91h:14039

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

Mark L. Green
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90024

Received by editor(s): June 5, 1999
Published electronically: November 29, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society