Suppose all my varieties are complex threefolds $X\rightarrow Y$ over some smooth base curve germ $Y$. We can assume the fibres are Del Pezzo surfaces with generic smooth fibre.


If I do (relative) log mmp over a klt pair $(X,D)$ what I obtain is a klt pair $(X',D')$ as output.


If I do (relative) mmp over a terminal $X$, I obtain a terminal $X'$ as output.


Now suppose $(X,D)$ is terminal. I would like to apply (relative) log mmp over it to obtain $(X',D')$ **terminal** as output. Of course this will not happen in general, but I was wondering if there are:


1) sufficient conditions on $(X,D)$ for this to happen, or


2) whether there is/may be some freedom when running MMP to make a choice in some/all the steps to guarantee this.


Probably the answer to both questions is NO, or even worse: 'probably not. I thought I would give it a try, anyway. It is kind of too specific to find it out of a book without learning all MMP (which I am actually trying to do at the same time).



Controlling singularities on log mmp
If $(X,D)$ is terminal and the stable base locus of $K\_X+D$ contains no components of the support of $D$, then any sequence of steps $f:X\to X'$ of the $K\_X+D$ MMP yields a terminal pair $(X',D'=f\_\*D)$. (Let $E$ be a divisor over $X'$ with discrepancy $a\_E(X',D')\leq 0$, then $a\_E(X,D)\leq a\_E(X',D')\leq 0$ so that $E$ is a divisor on $X$ which is contracted by $f$. But then $E$ is contained in the base locus of $K\_X+D$ and $E$ is contracted so that $a\_E(X,D)<a\_E(X',D')\leq 0$. But then $E$ is contained in the support of $D$.)


If the stable base locus contains components of the support of $D$, then it could still happen that $(X',D')$ is terminal, but if it is not one can still consider the terminalization of $(X',D')$. Roughly speaking this is similar to replacing $(X,D)$ by $(X,G)$ where $G$ is obtained by subtracting the common components of $D$ and the stable base locus of $K\_X+D$ from $D$. We then run the $K\_X+G$ MMP. Note that $H^0(m(K\_X+D))=H^0(m(K\_X+G))$.


