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. [Does the (torsion) Zariski cohomology of a (singular) hyperplane section of a smooth projective variety vanish (in small degrees)?](https://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.



Which (reducible) projective varieties could be presented as 'relatively smooth' hyperplane sections of irreducible (normal) ones?
Any *normal* projective scheme appears that way. Let $Y$ be normal projective and consider an embedding $Y\subseteq \mathbb P^n$ given by a *complete* linear system. Or more generally an embedding such that $Y\subseteq \mathbb P^n$ is projectively normal. (For the fact that this is indeed more general see [Hartshorne, Ex.II.5.14].) Projective normality implies that the projective cone $Z$ over $Y$ in $\mathbb P^{n+1}$ is normal and $Y$ is a hyperplane section of this cone.


For non-normal schemes it is not that clear-cut. For one thing you definitely need that it is $S\_1$, but if you only consider varieties, then this follows. 




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I wrote the above before Mikhail specified that he was interested in *reducible* $Y$'s. Here is an idea that might work for those:


Embed $Y$ in a projective space $\mathbb P^n$ in some way and take the projective cone $W$ over $Y$. Choose the coordinates so that $Y=W\cap H$ where $H=(x\_0=0)$. Now take the ideal of $W$ and in every polynomial in it replace each variable $x\_i$ ($i\neq 0$) with $x\_i+\varepsilon\_ix\_0$ for some not-yet-specified $\varepsilon\_i\in \mathbb C$ (but the $\varepsilon\_i$ should not depend on the polynomial) and call the variety defined by these equations $W\_\varepsilon$. In other words deform $W$ with $Y$ fixed. My guess is that in many cases $W\_\varepsilon$ will be normal for some general choice of $\varepsilon\_\bullet$. It should work at least when $Y$ is a complete intersection.


