From a fan to a toric variety

(The lattices)

For the torus $T = \GG_m^n$, the character lattice is $M = \Hom(T, \GG_m) \cong \ZZ^n$, with $\chi^m(t) = \prod t_i^{m_i}$, and the cocharacter lattice is $N = \Hom(\GG_m, T)$. The pairing $\inp{m}{u}$ is the integer $\ell$ with $\chi^m \circ \lambda^u : t \mapsto t^\ell$.

(Cones and fans)

A cone $\sigma \subseteq N_\RR$ is strongly convex rational polyhedral if it is generated by finitely many lattice vectors and contains no line. Its dual is $\dualof{\sigma} = \ts{m \in M_\RR \st \inp{m}{u} \geq 0 \text{ for all } u \in \sigma}$. A fan $\Sigma$ is a finite collection of such cones closed under faces, in which any two cones meet in a face of each.

(Cones with many edges)

A cone in $\RR^3$ can have any number of edges; the cross-sections in the figure have five edges $v_1, \ldots, v_5$ and four edges $v_1, \ldots, v_4$.

Two three-dimensional cones, with five and with four edges.

(The projective line)

The fan $\ts{\RR_{\geq 0}, \ts{0}, \RR_{\leq 0}}$ in $N_\RR = \RR$ gives two copies of $\CC$ glued along $\CC^\times$ by $x \mapsto x^{-1}$:

so $X_\Sigma = \PP^1$.

(The variety)

\[ \sigma \rightsquigarrow S_\sigma = \dualof{\sigma} \intersect M \rightsquigarrow U_\sigma = \Spec k[S_\sigma] , \] and $X_\Sigma$ is obtained by gluing the $U_\sigma$ along the $U_{\sigma \intersect \tau}$.

By Gordan’s lemma, $S_\sigma$ is a finitely generated semigroup, so $k[S_\sigma]$ is a finitely generated algebra and $U_\sigma$ is an affine variety.

The cone $\ts{0}$ gives the torus itself, the cone spanned by a basis gives $\AA^n$, and the fan of $\PP^n$ has rays $e_1,\ldots,e_n$ and $-\sum e_i$. For $d\geq1$, the cone spanned by $(0,1)$ and $(d,-1)$ gives the affine cone over the rational normal curve of degree $d$.

What are the character and cocharacter lattices?

What is a strongly convex rational polyhedral cone?

How is an affine toric variety built from a cone, and a toric variety from a fan?

Dependencies

D-VKR54$\Spec A$ as a locally ringed spaceuses

Backlinks

PR-FULLOCA face gives a principal open subset, which is how the affine charts glueuses
PR-FULSEPSeparation for convex polyhedral cones, and the double dualuses
PR-TORSMAFFSmooth affine toric varieties and why toric varieties are normaluses
FE-TORDUALThe affine toric variety of the cone on (2,1) and (1,4)uses
PR-O8V3IThe orbit-cone correspondenceuses
D-TORPOLYThe toric variety of a polytope and the normal fanuses

What links to this

The fan dictionary