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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).

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up vote 3 down vote accepted

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 $H^0(m(K_X+D))=H^0(m(K_X+G))$.

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Thanks Christopher! – Jesus Martinez Garcia Mar 17 at 22:48

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