Let $X$ be a projective variety and let $C_1,\ldots,C_n\subseteq X$ be some integral curves contained in $X$. Let $D$ be a divisor such that $D.C\geq 0$ for all curves $C\notin\{C_1,\ldots,C_n\}$. Then is it true that $D^d.V\geq 0$ for all $d$-dimensional integral subvarieties $V\subseteq X$ such that $V\not\subseteq C_1\cup\cdots\cup C_n$?
This seems to be pretty much equivalent to Kleiman's theorem for quasi-projective varieties. However, I have no idea whether it holds or not. How could one proceed?