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.[ref]
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