Adjunction and singular loci of hyperplane sections

Abstract

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