The module of logarithmic p-forms of a locally free arrangement
by
H. Schenck
Northeastern University

For an essential, central arrangement A subset or equal V =~ Kⁿ⁺¹ (char K = 0), let J be the Jacobian ideal of the defining polynomial of A. We show that \Omega¹(A) (the module of logarithmic one forms with pole along A) gives rise to a locally free sheaf on Pⁿ iff the sheaf Extⁱ_OPⁿ(O_Pⁿ/J, O_Pⁿ) is zero, for all i > 2, and prove that this condition holds if and only if for all X in L_A with rank X < dim V, the subarrangement A_X is free. \Omega¹(A) has a direct sum decomposition as \Omega¹₀ \oplusS(1); we prove that the sheaf defined by \Omega¹(A) is locally free if and only if the module of logarithmic p-forms \Omega^p(A) and the module \Lambda^p (\Omega¹(A)) have the same sheafification (so as modules, have isomorphic saturations). In this situation, we show that if either \Omega¹(A) or its dual has projective dimension at most one, then (writing c_t for the Chern polynomial) (1+t) ·c_t([(\Omega)\tilde]¹₀) = \Pi(A, t). The class of locally free arrangements for which this condition holds includes free arrangements, arrangements in P², and generic arrangements, and this inclusion is proper.

http://math.neu.edu/~schenck

Date received: April 26, 1999