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  "body": "\n## Definition\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]], let $n \\ge 0$, and let $\\mathbb{P}^n$ be the [[projective space|ZT9A6JK8QF]] over $k$.\nFor a set $T$ of [[homogeneous polynomials|2C3PAQ7H50]] in $k[x_0, \\dots, x_n]$, let\n$$V(T) = \\set{[a] \\in \\mathbb{P}^n : f(a) = 0 \\text{ for all } f \\in T}.$$\nThis is well defined, because $f(\\lambda a) = \\lambda^d f(a)$ when $f$ is homogeneous of degree $d$.\nA *projective algebraic set* is a set of the form $V(T)$; these are the closed sets of the Zariski topology on $\\mathbb{P}^n$.\nA *projective variety* is a projective algebraic set that is [[irreducible|SY2MWC434Z]] in this topology.\n\n## Remarks\n\nThe ideal of a projective algebraic set $X$ is the [[homogeneous ideal|6ZBNGS0P98]] generated by the homogeneous polynomials that vanish on $X$.\nThen $X$ is irreducible if and only if this ideal is prime.\n\nExample: the plane conic $C = V(xz - y^2) \\subseteq \\mathbb{P}^2$ is a projective variety.\nIts polynomial $xz - y^2$ is irreducible, so it generates a prime ideal, and $C$ is the image of $\\mathbb{P}^1$ under $[s : t] \\mapsto [s^2 : st : t^2]$.\n\nNon-example: $V(xy) \\subseteq \\mathbb{P}^2$ is a projective algebraic set but not a projective variety.\nIt is the union of the two lines $V(x)$ and $V(y)$.\n\nThe intersection of $X$ with a chart $U_i = \\set{x_i \\ne 0} \\cong k^n$ is an [[algebraic set|VHFMQQYSTN]] in $k^n$, defined by setting $x_i = 1$ in the equations; when nonempty, it is an [[affine variety|JERVQT4C12]].\nSome authors do not require irreducibility.\n",
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