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 -