= Toric geometry
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<Toric geometry> studies <algebraic varieties> with an action of an <algebraic torus> having a dense open orbit isomorphic to that <algebraic torus>. For normal <algebraic varieties> with a split <algebraic torus>, <toric fans> encode the geometry using integral <toric cones> and their intersections.
The simplest chart is the <affine line>: the multiplicative <algebraic torus> acts by scaling, with one open orbit and the origin as a fixed <point>. The <projective line> adds another fixed <point> at infinity. Its <toric fan> has the zero <toric cone> and the positive and negative half-lines in the real extension of the rank-one <toric lattice>.
The dictionary turns geometric questions into finite arithmetic data: <smoothness criterion for a toric variety> is checked by primitive <toric ray> generators extending to a <lattice basis>, properness by a <complete fan>, and <line bundles> by compatible integral linear <functions> on <toric cones>. The <normal variety> and <split algebraic torus> conventions matter; an arbitrary <algebraic variety> with a dense <algebraic torus> need not be described by the same ordinary <toric fan> construction.<Toric Varieties by David A. Cox and others>[chapters1–3, normal affine charts and fans]{ref}
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