Are there any restrictions known on a (complex reducible) projective variety $Y$ that can be presented as $Z\cap H$, where $Z$ is a normal (or just irreducible) closed subvariety of a smooth projective $P$, and $H$ is a smooth hyperplane section of $P$ ($Y$ is fixed, and everything else varies)? I am mostly interested in some cohomological obstructions here. The problem is that $Z$ is not necessarily smooth, so I don't know of any version of Weak Lefschetz that could be applied here. Should I look at some weights? cf. http://mathoverflow.net/questions/82432/does-the-torsion-zariski-cohomology-of-a-singular-hyperplane-section-of-a-smo
I am also interested in examples; yet I would prefer the dimension of $Y$ to be $>2$.
Upd. My idea was that a hyperplane section of $Z$ is often 'worse' than $Z$. From this point of view, it would be interesting to consider $Y$ that is not normal or even reducible (I didn't say about this in the first version of my question; sorry). In particular, I am interested in the case when $Y$ is a normal crossing scheme.