# Why are quasi-projective varieties locally affine?

From Wikipedia on Quasi Projective Varieties, it is stated that, "Quasi-projective varieties are locally affine in the same sense that a manifold is locally Euclidean : every point of a quasi-projective variety has a neighborhood which is an affine variety. This yields a basis of affine sets for the Zariski topology on a quasi-projective variety."

My question is, how can we explicitly show the above statement? The projective space $$P^n$$ can be covered by n affine space $$A^n$$ so do we just intersect these sets? And does this give us an open neighborhood that is still in the projective variety?

• Yes, that's the right idea. Why don't you try to work through it and see where you get stuck? Commented Dec 7, 2022 at 18:23
• We need (n+1) A^n to cover P^n.
– Yos
Commented Dec 7, 2022 at 18:24

You can justify this in two steps, which I'll leave you to try and show. We may write a quasiprojective variety as $$U \cap Z \subset \mathbb{P}^n$$ where $$U$$ is open and $$Z$$ is closed.
Step 0: $$\mathbb{A}^n$$ has a basis of affine open sets.
Step 1: $$\mathbb{P}^n$$ has a basis of affine open sets, so we may write $$U$$ as a union of affine open sets.
Now we can conclude every element of $$U \cap Z$$ has an affine open neighborhood.
Challenging exercise: If $$U, U'$$ are affine open subsets of a quasiprojective variety then $$U \cap U'$$ is affine.