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Hilbert schemes and Betti numbers over Clements–Lindström rings

Part of: Families, fibrations Commutative algebra: Homological methods

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
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Abstract

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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

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