The strong and weak Gamma-effective adjunction conjectures

Let $(X,D)$ be an klt (respectively, lc) pair with dimension $d$, and let $f:(X,D)\to Z$ be an lc-trivial fibration. An $\mathbb R$-b-divisor ${\bf M}$ is $\Gamma$-base-point free if it is a convex combination of base-point free b-divisors with coefficients in a finite set $\Gamma\subset(0,1]$. Let ${\bf D}_{\mo}$ denote the moduli b-divisor. Strong and weak $\Gamma$-effective adjunction conjectures. (1) If the coefficients of the horizontal divisors of $D$ belong to a finite set $I_h$, then there exist a finite set $\Gamma\subset(0,1]$ and a positive integer $m$, depending only on $d$ and $I_h$, such that $m{\bf D}_{\mo}$ is $\Gamma$-base-point free. (2) If the coefficients of $D$ belong to a DCC set $I\subset(0,1]$, then there exist a DCC set $J\subset(0,1]$, a finite set $\Gamma\subset(0,1]$, and a positive integer $m$, all depending only on $d$ and $I$, together with b-divisors $\widetilde{\bf D}_{\di}$ and $\widetilde{\bf D}_{\mo}$ satisfying: $(\widetilde{\bf D}_{\di})_Z\in J$; $m\widetilde{\bf D}_{\mo}$ is $\Gamma$-base-point free; $(Z,\widetilde{\bf D}_{\di})$ is klt (respectively, lc); and for birational morphisms $p:Z'\to Z$, $q:X'\to X$ and a morphism $f':X'\to Z'$ with $f\circ q=p\circ f'$, one has

$$q^*(K_X+D)\sim_{\mathbb R}f'^*(K_{Z'}+(\widetilde{\bf D}_{\di})_{Z'}+(\widetilde{\bf D}_{\mo})_{Z'}).$$

This variant is proposed to make effective adjunction compatible with inductions and real coefficients. Its status is open in the supplied source.