### 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 |

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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