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Classically, the natural map $\mathbb{A}_\mathbb{C}^{n+1}-\{O\} \rightarrow \mathbb{P}^n_\mathbb{C}$ maps every point in $\mathbb{A}_\mathbb{C}^{n+1}-\{O\}$ to the affine line that contains it. For arbitrary $k$, there is also a natural map from $\mathbb{A}_k^{n+1}-\{O\}$ to $\mathbb{P}^n_k$, which could be defined by considering the map for different affine opens as $D(x_i)$ and it is answered in the following question.

On a certain morphism of schemes from affine space to projective space.

So I have the question that is there a way to visualize this map like the classical one? I.e. given a prime ideal $\mathfrak{p}\subset k[x_0,\cdots,x_n]$, is there a straightforward way to find the homogeneous ideal it maps to?

For $k$-valued point in $\mathbb{A}_k^{n+1}-\{O\}$, is the answer the same as classical one? What can I state about other points?

edit: I think the image of $p$ is the largest homogeneous ideal contained in $p$.

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Wenzhe
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  • It's just the homogenization of the ideal. – Alex Youcis Mar 21 '16 at 23:09
  • If $I$ is an ideal in $k[x_1,\cdots,x_n]$, I know how to get its homogenization in $k[x_0,x_1,\cdots,x_n]$. But if $I$ is an ideal already in $k[x_0,x_1,\cdots,x_n]$, how could we get its homogenization? – Wenzhe Mar 21 '16 at 23:15
  • As in the question you've mentioned, you need and open cover of $\mathbb{P}^n$. On an affine chart it makes sense to see where ideals go... – Alan Muniz Mar 22 '16 at 01:16

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