# Log canonical centers of toric (and toroidal) varieties

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

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"?

• 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. Feb 17, 2020 at 17:56
• 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? Feb 18, 2020 at 2:18