Deletion–restriction for subspace arrangements

Let A be a subspace arrangement in V with a designated maximal affine subspace A 0 . Let A ′ = A ∖ { A 0 } be the deletion of A 0 from A and A ″ = { A ∩ A 0 | A ∩ A 0 ≠ ∅ } be the restriction of A to A 0 . Let M = V ∖ ⋃ A ∈ A A 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 → H k ( M ′ ) → H k ( M ) → H k + 1 − codim R ( A 0 ) ( 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 H k ( M ) for hyperplane arrangements to deletion–restriction subspace arrangements.