## Abstract

For an essential, central hyperplane arrangement A⊆V≃k^{n+1} we show that Ω^{1}(A) (the module of logarithmic one forms with poles along A) gives rise to a locally free sheaf on P^{n} if and only if, for all X∈L_{A} with rank X<dimV, the module Ω^{1}(A_{X}) is free. Motivated by a result of L. Solomon and H. Terao (1987, Adv. Math.64, 305-325), we give a formula for the Chern polynomial of a bundle E on P^{n} in terms of the Hilbert series of ⊕_{m∈Z}H^{0}(P^{n},∧^{i}E(m)). As a corollary, we prove that if the sheaf associated to Ω^{1}(A) is locally free, then π(A,t) is essentially the Chern polynomial. If Ω^{1}(A) has projective dimension one and is locally free, we give a minimal free resolution for Ω^{p} and show that Λ^{p}Ω^{1}(A)≃Ω^{p}(A), generalizing results of L. Rose and H. Terao (1991, J. Algebra136, 376-400) on generic arrangements.

Original language | English (US) |
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Pages (from-to) | 699-719 |

Number of pages | 21 |

Journal | Journal of Algebra |

Volume | 241 |

Issue number | 2 |

DOIs | |

State | Published - Jul 15 2001 |

## Keywords

- Chern polynomial
- Free resolution
- Hyperplane arrangement
- Poincaré polynomial
- Vector bundle

## ASJC Scopus subject areas

- Algebra and Number Theory