In algebraic geometry, a toric variety or torus embedding is a kind of algebraic variety that contains an algebraic torus whose group action extends to the whole variety. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special but still quite general class of toric varieties, this information is also encoded in a convex polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space.
Toric geometry in the context of Discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.
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