The information encoded in initial ideals

Fløystad, Gunnar; Green, Mark

doi:10.1090/S0002-9947-00-02737-9

The information encoded in initial ideals
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by Gunnar Fløystad and Mark L. Green PDF

Trans. Amer. Math. Soc. 353 (2001), 1427-1453 Request permission

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

David Bayer and Michael Stillman, A criterion for detecting $m$-regularity, Invent. Math. 87 (1987), no. 1, 1–11. MR 862710, DOI 10.1007/BF01389151
C. v. Bothmer. Thesis, University of Bayreuth (1995).
Michele Cook, The connectedness of space curve invariants, Compositio Math. 111 (1998), no. 2, 221–244. MR 1606169, DOI 10.1023/A:1000316500235
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Gunnar Fløystad, Higher initial ideals of homogeneous ideals, Mem. Amer. Math. Soc. 134 (1998), no. 638, viii+68. MR 1432143, DOI 10.1090/memo/0638
Gunnar Fløystad, A property deducible from the generic initial ideal, J. Pure Appl. Algebra 136 (1999), no. 2, 127–140. MR 1674773, DOI 10.1016/S0022-4049(97)00165-5
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), Lecture Notes in Math., Vol. 409, Springer, Berlin, 1974, pp. 543–579 (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
M. Green. Generic initial ideals. Summer school on commutative algebra, Barcelona 16. - 26. July 1996. Accompanying notes, volume II.
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
R. Liebling. Classification of space curves using initial ideals. Ph.D. thesis, University of California at Berkeley, (1996).
Mireille Martin-Deschamps and Daniel Perrin, Sur la classification des courbes gauches, Astérisque 184-185 (1990), 208 (French). MR 1073438