Let $(X,\Delta)$ be a klt pair and $D $ a $Q $-Cartier divisor on $X $ such that the ring of sections of $D $ is finitely generated. Let $c$ be the log canonical threshold of the asymptotic linear system $||D||$, and denote by $V=V (\mathcal{J}(c \cdot ||D||))$, consider $E $ a general $Q $-divisor on $||D||$ and $c'$ its lc threshold with respect to $(X,\Delta) $. Is $V $ a minimal lc center of $(X,\Delta+c'E) $?
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$\begingroup$ Not sure if I understand the question completely, but if $D$ is ample, then $\mathcal J (c\cdot ||D||)=\mathcal O _X$ for all $c\geq 0$ (actually you probably mean something like $\mathcal J (X,\Delta ; c\cdot ||D||)=\mathcal O _X$). On the other hand, for any $E\in |D|_Q$ we have $\mathcal J (c'\cdot |E|)\ne \mathcal O _X$ for all $c'\gg 0$. $\endgroup$– HaconCommented Oct 10, 2017 at 13:10
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