The module of logarithmic p-forms of a locally free arrangement

Mircea Mustaţǎ, Henry K. Schenck

Research output: Contribution to journalArticlepeer-review

Abstract

For an essential, central hyperplane arrangement A⊆V≃kn+1 we show that Ω1(A) (the module of logarithmic one forms with poles along A) gives rise to a locally free sheaf on Pn if and only if, for all X∈LA with rank X<dimV, the module Ω1(AX) 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 Pn in terms of the Hilbert series of ⊕m∈ZH0(Pn,∧iE(m)). As a corollary, we prove that if the sheaf associated to Ω1(A) is locally free, then π(A,t) is essentially the Chern polynomial. If Ω1(A) has projective dimension one and is locally free, we give a minimal free resolution for Ωp and show that ΛpΩ1(A)≃Ωp(A), generalizing results of L. Rose and H. Terao (1991, J. Algebra136, 376-400) on generic arrangements.

Original languageEnglish (US)
Pages (from-to)699-719
Number of pages21
JournalJournal of Algebra
Volume241
Issue number2
DOIs
StatePublished - Jul 15 2001

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

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