And is there a reference to which someone could point me for this?



Is the intersection of n-1 nef divisors in an n-dimensional algebraic variety guaranteed to be in the Mori cone?
The Mori cone $\overline{NE}(X)$ is the dual of $Nef(X)$. Let $C = D\_{1}\cap...\cap D\_{n-1}$ be the intersection of $n-1$ nef divisors. Since $Nef(X) = \overline{Amp}(X)$ we can find a sequence of divisors $D\_{i,\epsilon}$ with $\epsilon > 0$ such that $D\_{i,\epsilon}$ is ample for any $\epsilon >0$, and $D\_{i,\epsilon}\rightarrow D\_i$ for $\epsilon\mapsto 0$, for any $i = 1,...,n-1$. 


Now, $C\_{\epsilon} = D\_{1,\epsilon}\cap ...\cap D\_{n-1,\epsilon}$ is the intersection of $n-1$ ample divisors. Therefore $C\_{\epsilon}\in\overline{NE}(X)$ for any $\epsilon >0$, and $C\_{\epsilon}\rightarrow C$ for $\epsilon\mapsto 0$. Now, if $D$ is nef, we have $D\cdot C\_{\epsilon}\geq 0$ for any $\epsilon > 0$, because $\overline{NE}(X)$ is the dual of $Nef(X)$. This yields $ \lim\_{\epsilon \to 0}D\cdot C\_{\epsilon} = D\cdot C \geq 0$. That is $C\in\overline{NE}(X)$. 


