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Let $\mathbb{P}^n$ be a projective space over an algebraic closed field $k$. I want to show that any finitely many points in $\mathbb{P}^n$ can be included in an affine open chart of it. A candidate would be the principal open set i.e. the complement of a hypersurface $\mathbb{P}^n-V(f)$ which is affine. However, I have trouble constructing this homogeneous polynomial $f$ i.e. given finitely many points $a_0, a_1,...\in \mathbb{P}^n$ then $a_0,a_1,...\not\in V(f)$. The only thought I have is to look at them on an affine chart and I can construct an $f'$ in affine space, and then homogenize it. However, this will make me circular i.e. I come back to find an affine chart that includes those points. So I guess this won't work.

Appreciate any ideas or hints!

quuuuuin
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  • For a bit of a higher-level solution using prime avoidance (plus some good discussion), see here. There are also more basic solutions (linear algebra, good choices of coordinates, etc) if you're interested in those. Would you like me to mark this as a duplicate of the linked question? – KReiser Nov 02 '22 at 03:50
  • @KReiser Please feel free to do so. Would you mind pointing out the link for the elementary solutions as we haven't talked about scheme yet? – quuuuuin Nov 02 '22 at 14:42

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A fairly direct way to see this is via duality: we search for a hyperplane $H \subseteq \mathbb{P}^n_k$ such that $a_1,\dots,a_m \notin H$ - the complement of $H$ will thus contain $a_1,\dots,a_m$ and be isomorphic to $\mathbb{A}^n_k$.

If, by contradiction, every hyperplane $H$ contained $a_i$ for some $i$, then - passing to ${\mathbb{P}^n_k}^\ast \cong \mathbb{P}_k^n$ - we get that every point in $\mathbb{P}^n_k$ is contained in at least one of the $m$-many hyperplanes corresponding to $a_1,\dots,a_m \implies \mathbb{P}^n_k$ is a union of $m$ hyperplanes, which of course contradicts the fact the $\mathbb{P}^n_k$ has dimension $n > n-1$.

Thomas Manopulo
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