# ag.algebraic geometry – Distinguishing ample divisors by minimally intersecting curves on a projective simplicial toric variety

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

Does $$S(m) = S(m’)$$ imply that $$m = m’$$?

As a special case, assume that $$sigma$$ is the nef cone of a simplicial projective toric variety $$X_{Sigma}$$. Then my question seems to amount to the following:

If $$D_1$$ and $$D_2$$ are two ample divisors minimally generating in the ample cone, then does $$D_1 cdot C = 1 Leftrightarrow D_2 cdot C = 1$$ for all effective curves $$C$$ imply that $$D_1 = D_2$$?

This is the case I am most interested in.