|
Connectedness of Hilbert schemes
Author(s):
Irena
Peeva;
Mike
Stillman
Journal:
J. Algebraic Geom.
14
(2005),
193-211.
Posted:
October 26, 2004
Retrieve article in:
PDF
Abstract |
References |
Additional information
Abstract:
We show that the Hilbert scheme, that parametrizes all ideals with the same Hilbert function over an exterior algebra, is connected. We give a new proof of Hartshorne's Theorem that the classical Hilbert scheme is connected. More precisely: if  is either a polynomial ring or an exterior algebra, we prove that every two strongly stable ideals in  with the same Hilbert function are connected by a sequence of binomial Gröbner deformations.
References:
-
- [AAH]
- A. Aramova, L. Avramov, and J. Herzog: Resolutions of monomial ideals and cohomology over exterior algebras, Trans. Amer. Math. Soc. 352 (2000), 579-594. MR 1603874 (2000c:13021)
- [AHH]
- A. Aramova, J. Herzog, and T. Hibi: Gotzmann theorems for exterior algebras and combinatorics J. Algebra 191 (1997), 174-211. MR 1444495 (98c:13025)
- [Ei]
- D. Eisenbud: Commutative Algebra with a View Towards Algebraic Geometry, Springer Verlag, New York 1995. MR 1322960 (97a:13001)
- [EFS]
- D. Eisenbud, G. Fløystad, and F.-O. Schreyer: Sheaf cohomology and free resolutions over exterior algebras, Trans. Amer. Math. Soc. 355 (2003) 4397-4426. MR 1990756 (2004f:14031)
- [ESW]
- D. Eisenbud, F.-O. Schreyer, and J. Weyman: Resultants and Chow forms via Exterior Syzygies, J. Amer. Math. Soc. 16 (2003) 537-579. MR 1969204
- [EK]
- S. Eliahou and M. Kervaire: Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), 1-25. MR 1037391 (91b:13019)
- [Gr]
- A. Grothendieck: Techniques de construction et théorèmes d'existence en géométrie algébrique IV: Les schémas de Hilbert, Seminaire Bourbaki 13 (1960-61), #221.
- [Gr2]
- M. Green: Generic initial ideals, in Six lectures on commutative algebra, Birkhäuser, Progress in Mathematics 166 (1998), 119-185. MR 1648665 (99m:13040)
- [Ha]
- R. Hartshorne: Connectedness of the Hilbert scheme, Publications Mathématiques IHES 29 (1966), 5-48. MR 0213368 (35:4232)
- [Ka]
- G. Katona: A theorem for finite sets, Theory of Graphs (P. Erdös and G. Katona, eds.), Academic Press, New York (1968), 187-207. MR 0290982 (45:76)
- [Kr]
- J. Kruskal: The number of simplices in a complex, Mathematical Optimization Techniques (R. Bellman, ed.), University of California Press, Berkeley/Los Angeles (1963), 251-278. MR 0154827 (27:4771)
- [Ma]
- F. Macaulay: Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531-555.
- [Mac]
- D. Maclagan: Antichains of monomial ideals are finite, Proc. Amer. Math. Soc. 129 (2001), 1609-1615. MR 1814087 (2002f:13045)
- [Pa]
- K. Pardue: Deformation classes of graded modules and maximal Betti numbers, Ill. J. Math. 40 (1996), 564-585. MR 1415019 (97g:13029)
Additional Information:
Irena
Peeva
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853
Email:
irena@math.cornell.edu
Mike
Stillman
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853
PII:
S 1056-3911(04)00386-8
Received by editor(s):
March 26, 2003
Received by editor(s) in revised form:
January 4, 2004
Posted:
October 26, 2004
Additional Notes:
Both authors are partially supported by the NSF
|
