Divisors, class group and Picard group from the rays

Toric divisor theory is linear algebra on the rays: characters give the relations, the cokernel is the class group, and $K_X=-\sum_\rho D_\rho$.Suggest a slogan

(The two sequences)

Let $\Sigma$ have rays $\Sigma(1)$ with primitive generators $u_\rho$. Each ray $\rho$ gives a prime divisor $D_\rho$, the closure of the corresponding codimension-one orbit, and \[ \Div_T(X_\Sigma) = \bigoplus_{\rho \in \Sigma(1)} \ZZ D_\rho \cong \ZZ^{\Sigma(1)} . \] There are exact sequences \[ M \to \bigoplus_{\rho} \ZZ D_\rho \to \Cl(X_\Sigma) \to 0 , \qquad M \to \CDiv_T(X_\Sigma) \to \Pic(X_\Sigma) \to 0 , \] compatible with the inclusions $\CDiv_T \subseteq \Div_T$ and $\Pic \subseteq \Cl$. The left map is \[ m \mapsto \operatorname{div}(\chi^m) = \sum_{\rho \in \Sigma(1)} \inp{m}{u_\rho} D_\rho , \] whose kernel is $M \cap \bigcap_\rho u_\rho^\perp$. It is injective, so both sequences are short exact, exactly when the $u_\rho$ span $N_\RR$, that is, when $X_\Sigma$ has no torus factor. [@Ful93]

(Cartier divisors as a limit over the fan)

Order $\Sigma$ by inclusion of faces and let $M(\sigma) = \sigma^\perp \cap M$. Restriction to the affine opens $U_\sigma$ identifies \[ \CDiv_T(X_\Sigma) \cong \varprojlim_{\sigma \in \Sigma} M/M(\sigma) , \] the inverse limit over the face poset of the groups $\CDiv_T(U_\sigma) \cong M/M(\sigma)$ of local data $m_\sigma$.

(Quotient construction)

Suppose the $u_\rho$ span $N_\RR$. Let $S = \CC[x_\rho : \rho \in \Sigma(1)]$, for $\sigma \in \Sigma$ let $x^{\hat\sigma} = \prod_{\rho \notin \sigma(1)} x_\rho$, and let $B(\Sigma) = \gens{x^{\hat\sigma} : \sigma \in \Sigma}$ be the irrelevant ideal, generated by the $x^{\hat\sigma}$ for maximal $\sigma$. Then $Z(\Sigma) = V(B(\Sigma)) \subseteq \CC^{\Sigma(1)}$ is a union of coordinate subspaces of codimension at least $2$. Applying $\Hom_\ZZ(-, \CC^\times)$ to $0 \to M \to \ZZ^{\Sigma(1)} \to \Cl(X_\Sigma) \to 0$ gives \[ 1 \to G \to (\CC^\times)^{\Sigma(1)} \to T_N \to 1 , \qquad G = \Hom_\ZZ(\Cl(X_\Sigma), \CC^\times) , \] and $X_\Sigma$ is the good categorical quotient $\qty(\CC^{\Sigma(1)} \sm Z(\Sigma)) /\!\!/ G$. It is a geometric quotient if and only if $\Sigma$ is simplicial.

A primitive collection is a subset $C \subseteq \Sigma(1)$ not contained in $\sigma(1)$ for any $\sigma \in \Sigma$, every proper subset of which is contained in some $\sigma(1)$; then $Z(\Sigma) = \bigcup_C V(x_\rho : \rho \in C)$ over the primitive collections.

(The cone over the quadric surface)

Let $\sigma = \Cone(e_1, e_2, e_1 + e_3, e_2 + e_3) \subseteq \RR^3$, a cone with four edges, so not simplicial, and $X_\sigma = V(xz - yw) \subseteq \CC^4$, the affine cone over $\PP^1 \times \PP^1$.

A three-dimensional cone with four edges, which is not simplicial.

Here \[ \Cl(X_\sigma) = \operatorname{coker}\qty( \ZZ^3 \xrightarrow{A} \ZZ^4 ) = \ZZ , \qquad A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix} , \] generated by $D_\rho$ for any ray $\rho$, while $\Pic(X_\sigma) = 0$ because $X_\sigma$ is affine.

For $\PP^n$, with rays $e_1, \ldots, e_n$ and $e_0 = -\sum_i e_i$, the only primitive collection is $\ts{e_0, \ldots, e_n}$, so $Z = \ts{0}$, and $G = \CC^\times$ acts on $\CC^{n+1}$ by $\lambda \cdot (x_0, \ldots, x_n) = (\lambda x_0, \ldots, \lambda x_n)$, recovering $\PP^n = (\CC^{n+1} \sm \ts{0})/\CC^\times$.

(Canonical divisor)

$\omega_{X_\Sigma} \cong \OO\big( -\sum_\rho D_\rho \big)$, so $K_X \sim -\sum_{\rho \in \Sigma(1)} D_\rho$.

So the class group is the cokernel of an explicit integer matrix whose rows are the ray generators. Two consequences are immediate. It has rank $\size \Sigma(1) - \dim N_\RR$, and it has torsion exactly when the rays fail to generate $N$ as a lattice. When $\Sigma$ is simplicial, $\Pic$ has finite index in $\Cl$ and $\rank \Pic(X_\Sigma) = \size \Sigma(1) - \dim N_\RR$ as well — for $\FF_a$ this is $4 - 2 = 2$, a fibre and a section.

Cartier divisors are cut out locally: $D = \sum a_\rho D_\rho$ is Cartier iff for every maximal cone $\sigma$ there is $m_\sigma \in M$ with $\inp{m_\sigma}{u_\rho} = -a_\rho$ for all $\rho \leq \sigma$, and then $D|_{U_\sigma} = \operatorname{div}(\chi^{-m_\sigma})$. Equivalently $\CDiv_T(U_\sigma) \cong M/M(\sigma)$ where $M(\sigma) = \sigma^\perp \intersect M$.

For $\PP^n$ the $n+1$ rays give $D_0, \ldots, D_n$, all linearly equivalent, so $K_{\PP^n} \sim -(n+1)H$ and $\omega_{\PP^n} = \OO(-n-1)$ falls out of counting rays.

Describe the torus-invariant divisors on a toric variety and compute the class group.

What is the canonical divisor of a toric variety?

What is the rank of the Picard group of a simplicial toric variety?

For an integer matrix $V$, show that $\Hom_\ZZ(\operatorname{coker} V, \ZZ) \cong \ker V^t$.

For finitely generated free abelian groups $A, B$, show that $A^\vee \tensor B \cong \Hom(A, B)$.

How is a toric variety a GIT quotient, and when is that quotient geometric?

What is a primitive collection?

Dependencies

PR-O8V3IThe orbit-cone correspondenceuses

Backlinks

T-FULBETTIBetti numbers of a smooth complete toric variety from the cone countsuses
T-TORSURFSmooth complete toric surfaces and their intersection numbersuses
T-TORCOHOMThe Stanley--Reisner presentation of the cohomology of a smooth toric varietyuses
D-TORQDThe polytope of a divisor, and its lattice points as sectionsuses
FE-TORP2The projective plane, computed entirely from its fanuses
FE-TORWPSWeighted projective space as a toric varietyuses

What links to this

Polytopes and divisors