Let $X$ be a smooth, projective variety and $Y \subset X$ be a hyperplane section (possibly singular) of $X$. Suppose that the dimension of $X$ is $n$. Is it true that for any $k<n-1$, the induced morphism of intersection homologies: $$IH_k(Y) \to IH_k(X)$$ is an isomorphism?
I would think this is true as the version of the Lefschetz hyperplane section theorem I have it states this conclusion under the additional assumption that $Y$ is transverse to every strata of some Whitney stratification of $X$. As $X$ is smooth, in my case, I can take the only non-trivial strata to be $X$ and the other sub-strata to be empty sets. This should answer the above question, but I am not sure if I am making a mistake.