Theorem 12.25.14.F79036F002E1 Let $\theta $ be in the fixed cone, with parameters $a_ i,b_ i,c_ i,d_ i$, and let $D$ be a divisor of the above form. Assume
\[ a_ i\geq \delta _ i\text{ for all }i,\qquad c_ i\geq \rho _ i\text{ for every blown-up factor}. \]
Then the residual $\beta =\theta -\eta _ D$ lies in the fixed cone, and the common orders $a(\beta )$ and $a(\theta )$ satisfy
\[ a(\beta )-a(\theta )=\sum _ i\bigl (R_ i(\theta )-R_ i(\beta )\bigr )\geq 0. \]
Equality holds if and only if, for every factor,
\begin{align*} a_ i-\delta _ i+b_ i& >0& & \text{whenever }a_ i+b_ i>0,\\ c_ i-\rho _ i+d_ i& >0& & \text{whenever }a_ i=b_ i=0\text{ and }c_ i+d_ i>0; \end{align*}
the second condition concerns only blown-up factors.
Proof.
The residual parameters $(a_ i-\delta _ i,b_ i,c_ i-\rho _ i,d_ i)$ are nonnegative, so $\beta $ lies in the fixed cone; in particular it is smooth, semipositive and nef. Also
\[ (\theta -\beta )+\mathrm{dd}^{\mathrm{c}}\Phi _ D=[D]\geq 0, \]
which is the positive-current comparison of proposition 12.24.1. Apply theorem 12.25.8 to $\theta $ and to $\beta $. If $H_ i$ occurs in $\theta $ but not in $\beta $, the rank $R_ i$ drops from $m_ i$ to $r_ i<m_ i$ or to zero. If $H_ i$ does not occur in $\theta $, then $a_ i=b_ i=0$, and the range of lemma 12.25.13 together with $a_ i\geq \delta _ i$ forces $\delta _ i=0$ and hence $\rho _ i\geq 0$; thus $Q_ i$ cannot appear in $\beta $ if it was absent from $\theta $. Therefore no rank $R_ i$ increases, each difference $R_ i(\theta )-R_ i(\beta )$ is nonnegative, and the sum vanishes exactly when the generator that determines $R_ i(\theta )$ still occurs in $\beta $ for every $i$. This is the stated pair of conditions.
$\square$
Pome
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