Toric variety

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.

A precise definition is that a toric variety is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety.

Some authors also require it to be normal.

The original motivation to study toric varieties was to study torus embeddings. Given the algebraic torus ${\displaystyle T}$, the group of characters ${\displaystyle \hom(T,\mathbb {C} ^{*})}$ forms a lattice. Given a collection of points ${\displaystyle {\mathcal {A}}}$, a subset of this lattice, each point determines a map to ${\displaystyle \mathbb {C} ^{*}}$ and thus the collection determines a map to ${\displaystyle \left(\mathbb {C} ^{*}\right)^{|{\mathcal {A}}|}}$. By taking the Zariski closure of the image of such a map, one obtains an affine variety. If the collection of lattice points ${\displaystyle {\mathcal {A}}}$ generates the character lattice, this variety is a torus embedding. In similar fashion one may produce a parametrized projective toric variety, by taking the projective closure of the above map, viewing it as a map into an affine patch of projective space.

Given a projective toric variety, observe that we may probe its geometry by one-parameter subgroups. Each one parameter subgroup, determined by a point in the lattice, dual to the character lattice, is a punctured curve inside the projective toric variety. Since the variety is compact, this punctured curve has a unique limit point. Thus, by partitioning the one-parameter subgroup lattice by the limit points of punctured curves, we obtain a lattice fan, a collection of polyhedral rational cones. The cones of highest dimension correspond precisely to the torus fixed points, the limits of these punctured curves.

Suppose that ${\displaystyle N}$ is a finite-rank free abelian group, for instance the lattice ${\displaystyle \mathbb {Z} ^{n}}$, and let ${\displaystyle M}$ be its dual. A strongly convex rational polyhedral cone in ${\displaystyle N}$ is a convex cone (of the real vector space of ${\displaystyle N}$) with apex at the origin, generated by a finite number of vectors of ${\displaystyle N}$, and that contains no line through the origin. These will be called "cones" for short. When generated by a set of vectors ${\displaystyle v_{1},\dots ,v_{k}}$, it is denoted ${\displaystyle {\text{cone}}(v_{1},\ldots ,v_{k})=\left\{\sum _{i=1}^{k}a_{i}v_{i}\colon a_{i}\in \mathbb {R} _{\geq 0}\right\}}$. A one-dimensional cone is called a ray. For a cone ${\displaystyle \sigma }$, its affine toric variety ${\displaystyle U_{\sigma }}$ is the spectrum of the monoid algebra generated by the points of ${\displaystyle M}$ that are in the dual cone to ${\displaystyle \sigma }$.^{[further explanation needed]}

A (polyhedral) fan is a collection of (polyhedral) cones closed under taking intersections and faces. The underlying space of a fan ${\displaystyle \Sigma }$ is the union of its cones and is denoted by ${\displaystyle |\Sigma |}$.

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 ${\displaystyle U_{\sigma }}$ with an open subvariety of ${\displaystyle U_{\tau }}$ whenever ${\displaystyle \sigma }$ is a face of ${\displaystyle \tau }$. 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.