Negative curves of a big real divisor lie in finitely many divisors

Write a big real Cartier divisor on an integral projective variety as $D{\sim }_{\mathbb{R}}B+E$, with $B$ ample and $E$ effective real Cartier. Every curve outside $\mathrm{Supp}E$ has $D\cdot C=B\cdot C+E\cdot C>0$. Therefore curves negative against $D$ lie in finitely many support components of codimension one. On a projective scheme use componentwise bigness on a projective scheme and collect the supports on its finitely many integral components; codimension one is taken in the relevant component.

If a big class $D=B+E$ has ample restriction to every component $E_i$ of the effective support, then for any ample $A$ a sufficiently small $ϵ>0$ makes $D-ϵA$ nef. Use openness of the ample cone to keep $B-ϵA$ and all $(D-ϵA){|}_{E_i}$ ample with one finite minimum of bounds. Curves inside the support use these restrictions; curves outside use effectivity of $E$. Thus one does not need a uniform choice over infinitely many subvarieties.