Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 139 / 3 / v / a / 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 $j_i{D}_i\sim A+E_i$ with $E_i$ effective. ConsequentlyThe 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 (a) and (b) are equivalent.