the map on Picard groups induced by restriction to a toric subvariety

Question

Let $X$ be a (say, complex) toric variety acted upon by a torus $T$ and defined by a fan $\Sigma$ in the cocharacter lattice $N=\mathrm{Hom}(\mathbb{C}^\times, T)$, and let $M$ be the character lattice. For any cone $\sigma \in \Sigma$ put $M(\sigma) = \sigma^\perp \cap M$, $N(\sigma) = \mathrm{Hom}(M(\sigma), \mathbb{C}^\times)$. There is a natural projection $N \to N(\sigma)$. Then the closure of the orbit corresponding to $\sigma$ has the structure of a toric variety with respect to the quotient torus with the cocharacter lattice $N/N(\sigma)$ and given by the fan $Star(\sigma)$ consisting of the images in $N(\sigma)$ of the cones of $\Sigma$ containing $\sigma$. Note that the closed embedding $X_{Star(\sigma)} \to X$ is generally not a toric morphism, since the dense toric orbit of $X_{Star(\sigma)}$ does not intersect the dense toric orbit of $X$.

My question is: is there a way to describe the restriction map $\mathrm{Pic}(X) \to \mathrm{Pic}(X_{Star(\sigma)})$ in terms of the fans $\Sigma$ and $Star(\sigma)$?

Answer 1

A toric Cartier divisor $D$ is given by the Cartier data $\{m_\sigma\}_{\sigma \in \Sigma}$ [CLS, Theorem 4.2.8] where for each affine open chart $U_\sigma$, the toric coordinate $x^{-m_\sigma}$ is the equation for $D \cap U_\sigma$. One way to visualize Cartier data is to consider $\{m_\sigma\}$ as a piece-wise linear function on the support of the fan $|\Sigma| \subset N_\mathbf{R}$ [CLS, Theorem 4.2.12].

Adding a fixed $m \in M$ to each $m_\sigma$ does not change the linear equivalence class of $D$, and in fact $\mathrm{Pic}(X)$ consists of equivalence classes of the Cartier data.

To restrict $D$ to an orbit closure $V(\tau)$, we need to assume that $\mathrm{Supp}(D) \not\supset V(\tau)$. It follows from definitions that this condition is equivalent to $m_\tau = 0$, and subtracting $m_\tau$ from each component $m_\sigma$ we can assume that this is the case. Geometrically this means to move the divisor away from the given stratum.

Finally, under the assumption $m_\tau = 0$, the Cartier data $\{m_\sigma\}$ naturally restricts to Cartier data on $\mathrm{Star}(\tau)$! Indeed, in terms of piece-wise linear functions on $N$, the function given by $\{m_\sigma\}$, for $\sigma$ containing $\tau$ vanishes on $N(\sigma)$, and hence restricts to a piece-wise linear function on $N / N(\sigma)$.

References. [CLS] Cox-Little-Schenck "Toric Varieties"

Credits. This question was discussed in the Sheffield Algebraic Geometry Learning Seminar.