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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)?

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.

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If irreducible is fine, why not take $Z$ a projective cone over $Y$? –  algori Dec 28 '11 at 13:50
    
if $Y$ is reducible, isn't the corresponding cone reducible either? –  Mikhail Bondarko Dec 28 '11 at 20:26
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up vote 4 down vote accepted

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.


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.

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Thank you!! I will think about that. –  Mikhail Bondarko Dec 29 '11 at 8:49
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