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Given a log smooth pair $(X,B)$ with a reduced boundary divisor $B$, consider a birational model $\pi:X' \to X$ and a boundary divisor $B'$ which is given by $K_{X'}+B'=\pi^*(K_X+B)$. Here is my question:

(1) can we construct an example that $(X',B')$ is not sub-dlt?

(2) is there an lc center of $(X',B')$ which is not an irreducible component of an intersection locus of some certain components of $B'^{=1}$ (this means the coefficient one part of $B'$)?

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  • $\begingroup$ What about $\mathbb{A}^2$ with the two lines, and then you blow-up $(x,y^n)$? So I think the answer to the first one is yes. The second answer should be no, the morphism $X'\rightarrow X$ should be toroidal around the divisors with coefficient one. $\endgroup$
    – Joaquín Moraga
    Commented Feb 17, 2020 at 17:54
  • $\begingroup$ The second answer should be no, the morphism $X′\to X$ should be toroidal around the divisors with coefficient one. Thank you! This is exactly what I need. Could you please offer a reference or something? The term "toroidal" is really confusing to me.. $\endgroup$
    – Flyingpanda
    Commented Feb 18, 2020 at 2:28

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