For example, the complex projective plane CP2 may be represented by three complex coordinates satisfying
where the sum has been chosen to account for the real rescaling part of the projective map, and the coordinates must be moreover identified by the following action:
The approach of toric geometry is to write
The coordinates are non-negative, and they parameterize a triangle because
The triangle is the toric base of the complex projective plane. The generic fiber is a two-torus parameterized by the phases of ; the phase of can be chosen real and positive by the symmetry.
However, the two-torus degenerates into three different circles on the boundary of the triangle i.e. at or or because the phase of becomes inconsequential, respectively.
The precise orientation of the circles within the torus is usually depicted by the slope of the line intervals (the sides of the triangle, in this case).
Many more complicated complex manifolds, for example some del Pezzo surfaces, admit a toric description.
The origins of toric geometry were in particular compactification questions; but it was soon formulated as the geometric theory of algebraic varieties V defined by monomial sets of equations. The geometric equivalent to that is to have an action on V of an algebraic torus, with an open orbit. This is the theory of toric varieties or torus embeddings. Computationally they can be treated by means of the semigroup defined by the exponents in the monomials, making them particularly tractable.
Toric geometry can also be used in relation with invariant theory (particularly geometric invariant theory), roughly in the way maximal torus theory is applied to Lie groups, but relating to moduli spaces rather than representation theory.