Ramblings on quasi-projective varieties

This blog post is mean to be an exposition on quasi-projective varieties, something that I am having problems understanding. A quasi-projective variety is a locally closed projective variety. What does that mean? It means that it is the intesection of a Zariski open set and a Zariski closed set in some projective space. Does this align with what a locally closed set means? This article would confirm that this is indeed the case.

The wikipedia article on quasi-projective varieties states that an affine space is an open set in a projective space. How is this? I can surely understand that an affine space can be embedded in projective space. Then? Oh c’mon. Say the affine space $A^n$ is embedded as $U_i$ in $\Bbb{P}^n$ . The variety that lies in its complement is $z_i=0$ . Hence, being the complement of a closed set, the affine subspace is open.

As any affine variety in $\Bbb{A}^n$ can be embedded in $\Bbb{P}^n$ as the intersection of its topological closure in $\Bbb{P}^n$ and the affine chart $U_i$ , we see that any affine variety is a quasi-projective variety.