### 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 P^{r-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 language | English |
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Pages (from-to) | 155-172 |

Number of pages | 18 |

Journal | Mathematische Zeitschrift |

Volume | 267 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2011 Feb 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Zeitschrift*,

*267*(1), 155-172. https://doi.org/10.1007/s00209-009-0614-8

**Hilbert schemes and maximal Betti numbers over veronese rings.** / Gasharov, Vesselin; Murai, Satoshi; Peeva, Irena.

Research output: Contribution to journal › Article

*Mathematische Zeitschrift*, vol. 267, no. 1, pp. 155-172. https://doi.org/10.1007/s00209-009-0614-8

}

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.

UR - http://www.scopus.com/inward/record.url?scp=79551610671&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79551610671&partnerID=8YFLogxK

U2 - 10.1007/s00209-009-0614-8

DO - 10.1007/s00209-009-0614-8

M3 - Article

AN - SCOPUS:79551610671

VL - 267

SP - 155

EP - 172

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1

ER -