2000/01/30 by Mircea Mustata, Hal Schenck · 1 citation
Mathematics · #math.AG #msc:14J60 #msc:14Q10
published as J. Algebra 241 (2001), no. 2, 699-719. · LaTeX, 17 pages
arxiv created 2000/01/30 · arxiv updated 2009/11/30
For an essential, central hyperplane arrangement A in 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 in LA with rank X<dim V, the module Ω1(AX) is free. Our main result is that in this case the Poicare polynomial of A is essentially the Chern polynomial. The proof is based on a result of Solomon and Terao and on a formula we give for the Chern polynomial of a bundle E on Pn in terms of the Hilbert series of ⊕m H0(\wedgeiE(m)). If Ω1(A)has projective dimension one and is locally free, we give a minimal free resolution for Ωp, and show that \wedgep(Ω1(A))\isoΩp(A), generalizing results of Rose and Terao on generic arrangements.