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I'm having trouble with this question (it's a homework question). If $p:\mathbb{A}^{n+1}-\{0\} \rightarrow \mathbb{P}^n$ is the canonical projection and $Z\subset \mathbb{P}^n$ is closed, then $Z$ is irreducible if and only if $p^{-1}(Z)$ is irreducible.

I think I've got one direction "$\Leftarrow$": suppose $Z=C_{1}\cup C_{2}$, is a decomposition. Then $C_{i}=Z(I_{i})$ for some homogenous ideals $I_{i}$. Then $p^{-1}(C_{i})$ is the zero locus (in $\mathbb{A}^{n+1}$) of the ideal generated by all homogenous elements in $I_{i}$ without the origin. And this is closed in $\mathbb{A}^{n+1}-\{0\}$. Since $p^{-1}(Z)$ is irreducible then $p^{-1}(Z)=p^{-1}(C_{1})$, say. Then $Z=C_{1}$ is irreducible.

For the other implication I get stuck: suppose $p^{-1}(Z)=D_{1}\cup D_{2}$. Since $p$ is surjective $Z=p(D_{1})\cup p(D_{2})$. But I don't think $p(D_{i})$ are necessarily closed. I'm not sure how else to show irreducibility other than start with a decomposition, though.

Any help or hint is appreciated. Thanks in advance!

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Hint: Think about what is going on here. An irreducible subset of $\mathbb{P}^n$ is of the form $V_+(\mathfrak{p})$ where $\mathfrak{p}$ is a homogenous prime of $k[x_0,\ldots,x_n]$. The preimage is just $V(\mathfrak{p})$. So..

Alex Youcis
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