I am looking at the following theorem from "Toric Varieties" by Cox, Little and Schenk:

Theorem 6.4.11: If $X_{\Sigma}$ is a simplicial toric variety, then $\overline{NE}(X_{\Sigma}) = \large{\Sigma}_P \mathbf{R}_{\geq 0}r(P)$, where the sum is over all primitive collections $P$.

Such primitive collections $P$ gives rise to a 1-cycle in $N_1(X_{\Sigma})$ represented by an element $r(P) = (b_{\rho})_{\rho \in \Sigma(1)} \in \mathbf{R}^{\Sigma(1)}$, which can be shown to represent an effective class. Assuming that $X_{\Sigma}$ is a smooth projective variety, it is easy to show that the numbers $b_{\rho}$ are integral, in which case $r(P)$ sits inside $\overline{NE}(X_{\Sigma}) \cap \mathbf{Z}^n$.

Apparently, it is not known whether the $r(P)$'s always generate $\overline{NE}(X_{\Sigma}) \cap \mathbf{Z}^n$ as a semigroup.

However, I'm interested in the low-dimensional cases, specifically dimension 3.

My question is: Is it known, or can it be shown that the $r(P)$'s generate $\overline{NE}(X_{\Sigma}) \cap \mathbf{Z}^n$ for $X_{\Sigma}$ smooth, projective and of dimension $3$?


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