= Solution
An <ample real divisor> is a finite positive real combination of <ample Cartier divisors>:
$$
D=\sum_{j=1}^r a_jA_j,\qquad a_j>0.
$$
Equivalently its <numerical class> lies in the <ample cone>. This is a numerical condition even when the coefficients are irrational; it does not mean that some integer multiple of $D$ must be an integral divisor.
On an integral <projective variety>, a <big real divisor> is a finite positive real combination of <big Cartier divisors>. Equivalently, by the real form of <Kodaira's lemma>,
$$
\boxed{D\sim_{\mathbb R}B+E,\quad B\text{ ample real},\quad E\text{ effective real Cartier}.}
$$
For an integral <Cartier divisor>, bigness means maximal section-growth order $h^0(X,mD)\ge c m^{\dim X}$ along sufficiently divisible positive $m$, or <Iitaka dimension> $\dim X$. The <real linear equivalence of divisors> formulation permits finite positive combinations of effective <Cartier divisors>, whose supports are codimension one. The definitions and the ample-plus-effective formulation on integral varieties are discussed in https://www.math.kyoto-u.ac.jp/~fujino/big-r-divisor5.pdf[Fujino's notes on big real divisors].
For the paper's assertions on a general <projective scheme>, use <componentwise bigness on a projective scheme>: require bigness on every reduced irreducible component. All arguments below can then be carried out on those finitely many integral components; <ampleness> is also detected there. On a reducible scheme, merely asking for maximal total section growth on one component is insufficient. For example $X=\mathbb P^2\amalg\mathbb P^2$, with $D$ restricting to $\mathcal O(1)$ on the first component and $\mathcal O(-1)$ on the second, has quadratic total section growth, but negative intersection with every line in the second component. No finite collection of codimension-one <subvarieties> can contain all those lines. Thus that weaker meaning would make part (ii) false. In dimension zero the positivity statements are vacuous and every <line bundle> is <ample>; the compatible bigness convention also regards it as big.
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