Let (X,L) be a smooth polarized variety of dimension n. Let A ∈ |L| be an effective irreducible divisor, and let Σ be the singular locus of A. We assume that Σ is a smooth subvariety of dimension k ≥ 2, and codimension c ≥ 3, consisting of non-degenerate quadratic singularities. We study positivity conditions for adjoint bundles K_X+tL with t ≥ n−3. Several explicit examples motivate the discussion.

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