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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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. In other words, Or more generally an embedding such that $Y\subseteq \mathbb P^n$ is projectively normal. (For the fact that this implication 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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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. In other words, such that $Y\subseteq \mathbb P^n$ is projectively normal. (For this implication 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.