Quasi-projective variety

Remarks

Equivalently, $X$ is an irreducible subset of $\mathbb{P}^n$ of the form $Y \setminus Z$, where $Y$ and $Z$ are projective algebraic sets. A nonempty open subset of an irreducible space is irreducible and dense, so $X$ is irreducible and its closure is $Y$.

Example: every affine variety $X \subseteq k^n$ is quasi-projective. Identify $k^n$ with the chart $U_0 = \set{x_0 \ne 0}$ of $\mathbb{P}^n$; then $X$ is open in its closure $\overline{X} \subseteq \mathbb{P}^n$, which is a projective variety. Every projective variety is quasi-projective, taking $X = Y$.

Example: $k^2 \setminus \set{(0, 0)}$ is quasi-projective, as an open subset of $\mathbb{P}^2$; it is neither affine nor projective.

Non-example: $V(xy) \setminus \set{[0 : 0 : 1]} \subseteq \mathbb{P}^2$ is open in a projective algebraic set but is not a quasi-projective variety, because it is the union of two proper closed subsets, the two punctured lines.