Deletion–restriction for subspace arrangements

Let A be a subspace arrangement in V with a designated maximal affine subspace A0. Let A′=A∖{A0} be the deletion of A0 from A and A″={A∩A0|A∩A0≠∅} be the restriction of A to A0. Let M=V∖⋃A∈AA be the complement of A in V. If A is an arrangement of complex affine hyperplanes, then there is a split short exact sequence, 0→Hk(M′)→Hk(M)→Hk+1−codimR(A0)(M″)→0. In this paper, we determine conditions for when the triple (A,A′,A″) of arrangements of affine subspaces yields the above split short exact sequence. We then generalize the no-broken-circuit basis nbc of Hk(M) for hyperplane arrangements to deletion–restriction subspace arrangements.