= Toric variety
{wiki}
A <normal variety>[normal] split <toric variety> is an <algebraic variety> containing a <split algebraic torus> $T$ as a dense <open subset>, with the multiplication action of $T$ extended to the whole <algebraic variety>. The <toric fan> construction glues
$$
U_\sigma=\operatorname{Spec}k[\sigma^\vee\cap M]
$$
for <toric cones> of a <toric fan>, along the open charts corresponding to common faces. <Gordan lemma> makes each chart finitely generated, and saturation of the <toric lattice> monoid makes the chart a <normal variety>.
The first-quadrant <toric cone> gives <affine space> $\mathbb A^n$; the zero <toric cone> gives the <algebraic torus> $\mathbb G_m^n$. The two half-lines in rank one give the <projective line>. These examples show that a toric <algebraic variety> need not be complete or have any fixed <points>.
Some authors allow nonnormal toric <algebraic varieties> as well. Such <algebraic varieties> need not arise from ordinary <toric fans>; the restriction to <normal varieties> is essential in the stated <toric fan> classification.<Toric Varieties by David A. Cox and others>[chapters1 and3; author book page, nonnormal toric varieties]{ref}
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