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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field and let $\\mathbb{P}^n$ be the [[projective space|ZT9A6JK8QF]] over $k$ with the Zariski topology.\nA *quasi-projective variety* is a nonempty [[open subset|02BH3MCBQB]] $X$ of a [[projective variety|A4H4CZ2V35]] $Y \\subseteq \\mathbb{P}^n$, with the [[subspace topology|0WS9P12FQR]].\n\n## Remarks\n\nEquivalently, $X$ is an [[irreducible|SY2MWC434Z]] subset of $\\mathbb{P}^n$ of the form $Y \\setminus Z$, where $Y$ and $Z$ are projective algebraic sets.\nA nonempty open subset of an irreducible space is irreducible and dense, so $X$ is irreducible and its closure is $Y$.\n\nExample: every [[affine variety|JERVQT4C12]] $X \\subseteq k^n$ is quasi-projective.\nIdentify $k^n$ 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.\nEvery projective variety is quasi-projective, taking $X = Y$.\n\nExample: $k^2 \\setminus \\set{(0, 0)}$ is quasi-projective, as an open subset of $\\mathbb{P}^2$; it is neither affine nor projective.\n\nNon-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.\n",
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