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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 $Star(\sigma)$?

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  • $\begingroup$ In general, that is in the singular case, class groups do not admit pull-backs for closed embeddings: a typical problem is when the Weil divisor does not intersect the singular locus properly, e.g. contains the singular locus, one may not be able to move it off to make an intersection of the the correct dimension. $\endgroup$ – Evgeny Shinder Jan 20 at 9:21
  • $\begingroup$ fair point, I have changed the question to be about Picard groups $\endgroup$ – Dima Sustretov Jan 20 at 16:17
  • $\begingroup$ Just a comment. The Picard group has a description in terms of piecewise linear functions on the fan, and the pullback map should be compatible with pulling back these functions. $\endgroup$ – Piotr Achinger Jan 20 at 22:00
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    $\begingroup$ @PiotrAchinger: the problem is that the inclusion is not a toric morphism, so is not defined by a linear map of cocharacter lattices. It is an equivariant morphism (though it seems that there is no canonical section of the quotient projection, so you first have to pick one), but I am confused as to what combinatorial data defines it and how it can be used to describe the map on the Picard groups. $\endgroup$ – Dima Sustretov Jan 28 at 0:27
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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.

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