Hilbert schemes and maximal Betti numbers over veronese rings

Vesselin Gasharov, Satoshi Murai, Irena Peeva

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Abstract

Macaulay's Theorem (Macaulay in Proc. Lond Math Soc 26:531-555, 1927) characterizes the Hilbert functions of graded ideals in a polynomial ring over a field. We characterize the Hilbert functions of graded ideals in a Veronese ring R (the coordinate ring of a Veronese embedding of Pr-1). We also prove that the Hilbert scheme, which parametrizes all graded ideals in R with a fixed Hilbert function, is connected; this is an analogue of Hartshorne's Theorem (Hartshorne in Math. IHES 29:5-48, 1966) that Hilbert schemes over a polynomial ring are connected. Furthermore, we prove that each lex ideal in R has the greatest Betti numbers among all graded ideals with the same Hilbert function.

Original languageEnglish
Pages (from-to)155-172
Number of pages18
JournalMathematische Zeitschrift
Volume267
Issue number1
DOIs
Publication statusPublished - 2011 Feb 1

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ASJC Scopus subject areas

  • Mathematics(all)

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