# Distinguishing ample divisors by minimally intersecting curves on a smooth projective toric variety

My question has an easily formulated generalization, which I will state first. Let $$\sigma \subseteq \mathbf{R}^n$$ be a full-dimensional strongly convex polyhedral cone. For each lattice point $$m \in \sigma^o \cap \mathbf{Z}^n$$, minimally generating inside the interior cone $$\sigma^o$$, let $$S(m) \subseteq \sigma^{\vee} \cap \mathbf{Z}^n$$ denote the set of lattice points $$u$$ with $$\langle u,m \rangle = 1$$. The generalized question is:

Does $$S(m) = S(m') \not = \varnothing$$ imply that $$m = m'$$?

UPDATE: As Minseon Shin pointed out, there was a $$2$$-dimensional counter-example to the previous formulation of the above.

For my main question, as a special case of the above, assume that $$\sigma$$ is the nef cone of a smooth projective toric variety $$X_{\Sigma}$$. Then my question amounts to the following:

Let $$D_1$$ and $$D_2$$ be two ample divisors, minimally generating inside the ample cone. Suppose that there exists two effective curves $$C_1,C_2$$ such that $$D_1 \cdot C_1 = D_2 \cdot C_2 = 1$$. Then does $$D_1 \cdot C = 1 \Leftrightarrow D_2 \cdot C = 1$$ for all effective curves $$C$$ imply that $$D_1 = D_2$$?

• Is the following a counterexample to the generalization? Let $n = 2$, let $s \ge 4$ be an integer, let $\sigma \subseteq \mathbf{R}^{2}$ be the cone generated by $(s,1)$ and $(0,1)$, let $m = (1,1)$ and $m' = (2,1)$. Then $S(m) = S(m') = \{(0,1)\}$. – Minseon Shin Aug 4 at 23:47
• Yes, thanks - I'll update my question. – Mellon Aug 5 at 9:19