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Q1: Let $(X,B)$ be a toric variety. There exists a toric resolution of singularities $f:(Y,E) \to (X,B)$. Here is my question:

Is any lc center of $(X,B)$ an irreducible component of an intersection locus of some certain components of $B$?

Q2: Consider a toroidal variety $(X,B)$ (for definition, see [AK] [Kawamata]) https://arxiv.org/pdf/alg-geom/9707012.pdf

and https://arxiv.org/pdf/1008.1489.pdf

if $(X,B)$ is quasi-smooth (i.e.each cone $\sigma_x$ is simplicial, or each local toric model has only abelian quotient singularities.), then $X$ is $\mathbb{Q}$-factorial klt. Here is my question:

(1) according to [AK, pg.5], $(X,B)$ has a stratification by intersections of components of $B$. Does it imply all lc centers are strata of this structure?

(2) since each component of $B$ is required to be normal, how far is "quasi-smooth" from "dlt"?

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  • $\begingroup$ For log toric pairs dlt implies log smooth. $\mathbb{Q}$-factorial and dlt just implies finite quotient of a log smooth point. The answer of $(1)$ is yes. $\endgroup$
    – Joaquín Moraga
    Commented Feb 17, 2020 at 17:56
  • $\begingroup$ Thank you Joaquin! Where can I find a reference of (1)? So the singularities of a toric variety as a log pair are always like "intersecting not transversally" of boundary divisors? $\endgroup$
    – Flyingpanda
    Commented Feb 18, 2020 at 2:18

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