Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-13T04:47:59.583Z Has data issue: false hasContentIssue false
  • English
  • Français

Hilbert schemes and Betti numbers over Clements–Lindström rings

Part of: Commutative algebra: Homological methods Families, fibrations

Published online by Cambridge University Press:  25 July 2012

Satoshi Murai  and
Irena Peeva
Satoshi Murai
Affiliation:
Department of Mathematical Science, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan (email: murai@yamaguchi-u.ac.jp)
Irena Peeva
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the Hilbert scheme, that parameterizes all ideals with the same Hilbert function over a Clements–Lindström ring W, is connected. More precisely, we prove that every graded ideal is connected by a sequence of deformations to the lex-plus-powers ideal with the same Hilbert function. This is an analogue of Hartshorne’s theorem that Grothendieck’s Hilbert scheme is connected. We also prove a conjecture by Gasharov, Hibi, and Peeva that the lex ideal attains maximal Betti numbers among all graded ideals in W with a fixed Hilbert function.

Keywords

Hilbert schemedeformationsBetti numbers

MSC classification

Secondary: 14D15: Formal methods; deformations 14D20: Algebraic moduli problems, moduli of vector bundles 13D02: Syzygies, resolutions, complexes
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 5 , September 2012 , pp. 1337 - 1364
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[AHH98]Aramova, A., Herzog, J. and Hibi, T., Squarefree lexsegment ideals, Math. Z. 228 (1998), 353–378.CrossRefGoogle Scholar
[CL69]Clements, G. and Lindström, B., A generalization of a combinatorial theorem of Macaulay, J. Combin. Theory 7 (1969), 230–238.CrossRefGoogle Scholar
[Eis95]Eisenbud, D., Commutative algebra with a view towards algebraic geometry (Springer, New York, 1995).Google Scholar
[GHP02]Gasharov, V., Hibi, T. and Peeva, I., Resolutions of a-stable ideals, J. Algebra 254 (2002), 375–394.CrossRefGoogle Scholar
[GHP08]Gasharov, V., Horwitz, N. and Peeva, I., Hilbert functions over toric rings, Michigan Math. J. 57 (2008), 339–359. Special volume in honor of Melvin Hochster.CrossRefGoogle Scholar
[Gro60/61]Grothendieck, A., Techniques de construction et théorèmes d’existence en géométrie algébrique IV: Les schémas de Hilbert, Semin. Bourbaki 13 (1960–61), #221.Google Scholar
[Har66]Hartshorne, R., Connectedness of the Hilbert scheme, Publ. Math. Inst. Hautes Études Sci. 29 (1966), 5–48.CrossRefGoogle Scholar
[Mac27]Macaulay, F., Some properties of enumeration in the theory of modular systems, Proc. Lond. Math. Soc. 26 (1927), 531–555.CrossRefGoogle Scholar
[MM11]Mermin, J. and Murai, S., The lex-plus-powers conjecture holds for pure powers, Adv. Math. 226 (2011), 3511–3539.CrossRefGoogle Scholar
[Par96]Pardue, K., Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), 564–585.CrossRefGoogle Scholar
[Pee04]Peeva, I., Consecutive cancellations in Betti numbers, Proc. Amer. Math. Soc. 132 (2004), 3503–3507.CrossRefGoogle Scholar
[PS05]Peeva, I. and Stillman, M., Connectedness of Hilbert schemes, J. Algebraic Geom. 14 (2005), 193–211.CrossRefGoogle Scholar