Let X be a projective scheme over an artinian commutative ring $R_0$. Let $\Cal F$ be a coherent sheaf of $\Cal O_X$-modules. We present a sample of bounding results for the so called cohomological Hilbert functions

\centerline{$h^i_{X, \Cal F} : \Bbb Z \to \Bbb N_0, n \mapsto h^i_{X, \Cal F} (n) =

length_{R_0} H^i (X, \Cal F (n))$}

of $\Cal F$. Our main interest is to bound these functions in terms of the so called cohomology diagonal $(h^j_{X, \Cal F} (-j))^{\dim (\Cal F)}_{j = 0}$ of $\Cal F$. Our results present themselves as quantitative versions of the vanishing theorems of Castelnuovo-Serre and of Severi-Enriques-Zariski-Serre. In particular we get polynomial bounds for the (Castelnuovo) regularity at arbitrary levels and for the (Severi) coregularity at any level below the global subdepth $\delta (\Cal F) := \min{depth(\Cal F_x) | x \in X, x closed}$ of $\Cal F$.

We also show that the cohomology diagonal of $\Cal F$ provides minimal bounding systems for the mentioned regularities and coregularities.

As a fundamental tool we use an extended version of the method of linear systems of general hyperplane sections.

Brodmann, M; Matteotti, C; Nguyen, D M (2000). *Bounds for cohomological Hilbert-functions of projective schemes over Artinian rings.* Vietnam Journal of Mathematics, 28(4):341-380.

## Abstract

Let X be a projective scheme over an artinian commutative ring $R_0$. Let $\Cal F$ be a coherent sheaf of $\Cal O_X$-modules. We present a sample of bounding results for the so called cohomological Hilbert functions

\centerline{$h^i_{X, \Cal F} : \Bbb Z \to \Bbb N_0, n \mapsto h^i_{X, \Cal F} (n) =

length_{R_0} H^i (X, \Cal F (n))$}

of $\Cal F$. Our main interest is to bound these functions in terms of the so called cohomology diagonal $(h^j_{X, \Cal F} (-j))^{\dim (\Cal F)}_{j = 0}$ of $\Cal F$. Our results present themselves as quantitative versions of the vanishing theorems of Castelnuovo-Serre and of Severi-Enriques-Zariski-Serre. In particular we get polynomial bounds for the (Castelnuovo) regularity at arbitrary levels and for the (Severi) coregularity at any level below the global subdepth $\delta (\Cal F) := \min{depth(\Cal F_x) | x \in X, x closed}$ of $\Cal F$.

We also show that the cohomology diagonal of $\Cal F$ provides minimal bounding systems for the mentioned regularities and coregularities.

As a fundamental tool we use an extended version of the method of linear systems of general hyperplane sections.

## Citations

## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | cohomological Hilbert functions; Castelnuovo bounds; Severi bounds |

Language: | English |

Date: | 2000 |

Deposited On: | 31 May 2010 15:25 |

Last Modified: | 12 Aug 2016 12:11 |

Publisher: | Springer |

ISSN: | 2305-221X |

Official URL: | http://www.math.ac.vn/publications/vjm/vjm_28/341.html |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1810157 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1008.13004 |

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