Affine toric variety and polyhedral cone Suppose that N is a finite-rank
free abelian group, for instance the
lattice \mathbb{Z}^n, and let M be its dual. A strongly convex rational polyhedral cone in N is a
convex cone (of the real vector space of N) with apex at the origin, generated by a finite number of vectors of N, and that contains no line through the origin. These will be called "cones" for short. When generated by a set of vectors v_1,\dots,v_k, it is denoted \text{cone}(v_1,\ldots,v_k)=\left\{\sum_{i=1}^ka_i v_i\colon a_i\in\mathbb{R}_{\geq 0}\right\}. A one-dimensional cone is called a
ray. For a cone \sigma, its affine toric variety U_\sigma is the
spectrum of the
monoid algebra generated by the points of M that are in the
dual cone to \sigma.
Fundamental theorem for toric geometry A
(polyhedral) fan is a collection of (polyhedral)
cones closed under taking intersections and faces. The
underlying space of a fan \Sigma is the union of its cones and is denoted by |\Sigma|. The toric variety of a fan of strongly convex rational cones is given by taking the affine toric varieties of its cones and gluing them together by identifying U_\sigma with an open subvariety of U_\tau whenever \sigma is a face of \tau. The toric variety constructed from a fan is necessarily
normal. Conversely, every toric variety has an associated fan of strongly convex rational cones. This correspondence is called the
fundamental theorem for toric geometry, and it gives a one-to-one correspondence between normal toric varieties and fans of strongly convex rational cones. The fan associated with a toric variety condenses some important data about the variety. For example, the
Cartier divisors are associated to the rays of the fan. Moreover, a toric variety is
smooth, or
nonsingular, if every cone in its fan can be generated by a subset of a
basis for the free abelian group N, and it is
complete if its fan is complete, that is, its underlying space is the whole vector space.
Morphisms of toric varieties Suppose that \Sigma_1 and \Sigma_2 are fans in lattices N_1 and N_2, respectively. If f is a linear map from N_1 to N_2 such that the image of every cone of \Sigma_1 is contained in a cone of \Sigma_2, then f induces a morphism f_* between the corresponding toric varieties. This map f_* is
proper if and only if the preimage of |\Sigma_2| under the map f is |\Sigma_1|.
Projective toric variety, the ones coming from polytopes A toric variety is
projective if it can be embedded in some
complex projective space. Let P be a
polytope. For any vertex v of P, the
normal cone of P at vertex v is the cone generated by the
outer normals of the
facets containing v. The
normal fan of P is the fan whose maximal cones are the normal cones at each vertex of P. It is well known that projective toric varieties are the ones coming from the normal fans of rational polytopes. For example, the
complex projective plane \mathbb{CP}^2 comes from the triangle, or 2-
simplex. It may be represented by three complex coordinates satisfying :|z_1|^2+|z_2|^2+|z_3|^2 = 1 , \,\! 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
U(1) action: :(z_1,z_2,z_3)\approx e^{i\phi} (z_1,z_2,z_3) . \,\! The approach of toric geometry is to write :(x,y,z) = (|z_1|^2,|z_2|^2,|z_3|^2) . \,\! The coordinates x,y,z are non-negative, and they parameterize a triangle because :x+y+z=1 ; \,\! that is, :\quad z=1-x-y . \,\! The triangle is the
toric base of the complex projective plane. The generic fiber is a two-torus parameterized by the phases of z_1,z_2; the phase of z_3 can be chosen real and positive by the U(1) symmetry. However, the two-torus degenerates into three different circles on the boundary of the triangle i.e. at x=0 or y=0 or z=0 because the phase of z_1,z_2,z_3 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). Note that this construction is related to
symplectic geometry as the map \begin{cases}\mathbb{CP}^2&\to \mathbb{R}_{\geq 0}\\(z_1,z_2,z_3)&\mapsto |z_1|+|z_2|+|z_3|\end{cases} is related to the
moment map for the action of U(1) on the symplectic manifold \mathbb{CP}^2. ==Classification of smooth complete toric varieties==