Hilbert Schemes and Betti Numbers over a Clements-lindström Ring
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چکیده
We show that the Hilbert scheme, that parametrizes all ideals with the same Hilbert function over a Clements-Lindtröm 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. Our result is an analogue of Hartshorne’s theorem that Grothendieck’s Hilbert scheme is connected; however our proof is entirely different, since Hartshorne’s deformations (distractions) do not work over W. 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.
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تاریخ انتشار 2008