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  "body": "\n## Definition\n\nLet $k$ be a field, and let $T \\cong \\mathbb{G}_m^n$ be a split [[algebraic torus|66FQFAJNS4]] over $k$.\nA *toric variety* with torus $T$ is a [[variety|HV6YBS0YDH]] $X$ over $k$ with an open embedding $T \\hookrightarrow X$ such that:\n\n1. $X$ is [[normal|EK261G1XP9]];\n2. the image of $T$ is dense in $X$;\n3. the action of $T$ on itself by multiplication extends to an [[action|RW85C34GQW]] $T \\times X \\to X$.\n\n## Remarks\n\nSome authors drop clause 1 and speak of normal toric varieties when it holds.\nNormal toric varieties correspond to fans: finite collections of strongly convex rational polyhedral cones in $\\R^n$, closed under faces, any two of which meet in a common face.\n\nExample: $\\mathbb{P}^n$ with the torus $T = \\set{[x_0 : \\cdots : x_n] : x_0 x_1 \\cdots x_n \\ne 0} \\cong \\mathbb{G}_m^n$.\nHere $T$ is open and dense, $\\mathbb{P}^n$ is smooth and hence normal, and $(t_1, \\ldots, t_n) \\cdot [x_0 : x_1 : \\cdots : x_n] = [x_0 : t_1 x_1 : \\cdots : t_n x_n]$ extends the multiplication.\n\nNon-example: the cuspidal curve $C = \\set{y^2 = x^3}$ in $\\mathbb{A}^2$ contains the dense open torus $\\set{(t^2, t^3) : t \\ne 0}$.\nThe action $s \\cdot (x, y) = (s^2 x, s^3 y)$ extends the multiplication, so clauses 2 and 3 hold.\nBut $C$ is not normal at the origin, so it fails clause 1.\n",
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