TY - JOUR
T1 - Hilbert schemes and maximal Betti numbers over veronese rings
AU - Gasharov, Vesselin
AU - Murai, Satoshi
AU - Peeva, Irena
PY - 2011/2/1
Y1 - 2011/2/1
N2 - 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.
AB - 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.
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U2 - 10.1007/s00209-009-0614-8
DO - 10.1007/s00209-009-0614-8
M3 - Article
AN - SCOPUS:79551610671
SN - 0025-5874
VL - 267
SP - 155
EP - 172
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 1
ER -