= Solution
Condition (a) is the definition of a <big real divisor>. Fix an ample Cartier divisor $A$. For each big Cartier divisor $D_i$, part (iii) gives $j_iD_i\sim A+E_i$ with $E_i$ effective. Consequently
$$
D\sim_{\mathbb R}\left(\sum_i\frac{d_i}{j_i}\right)A+\sum_i\frac{d_i}{j_i}E_i.
$$
The first coefficient is positive, proving (b).
For the converse, suppose $D\sim_{\mathbb R}tA+E$, with $t>0$ and $E$ effective. Apply <rational approximation of an ample-plus-effective real divisor>: in a finite-dimensional space generated by Cartier divisors and the finitely many principal divisors occurring in this relation, express $D$ as a positive convex combination of rational divisors $B_\nu$, each satisfying $B_\nu\sim_{\mathbb Q}t_\nu A+E_\nu$ with $t_\nu>0$ and $E_\nu$ effective. After clearing denominators, part (iii) shows that each $B_\nu$ is a positive rational multiple of a big Cartier divisor. This gives the actual equality required by (a), rather than merely a numerical or linear equivalence.
The approximation lemma keeps the finitely many effectivity inequalities and linear-equivalence equations simultaneously; see its proof for the rational-face argument and the reduction of nonnormal varieties by finite normalization. Therefore \b[(a) and (b) are equivalent].
Back to article page
