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Let $X$ be a smooth toric projective variety. Let $T$ be the big torus acting on $X$. Let $D=X\backslash T$ be the boundary divisor.

Question 1. Will $D_i$ be a smooth toric projective variety for each irreducible component of $D$?

Question 2. Can we replace "smooth" by "normal" above?

P.S. Only over char=0 field.

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    $\begingroup$ Q1. Yes, as an affine toric variety $\operatorname{Spec} k[P]$ is smooth if and only if $P\simeq \mathbf{N}^r \times \mathbf{Z}^s$, and so it is easy to see that a smooth toric variety is locally of the form $X = \mathbf{A}^n$, $D=$ union of some coordinate hyperplaces. Q2. No, take a rational simplicial cone in $\mathbf{R}^3$ whose faces are not generated by their rays. I recommend consulting some basic reference on toric varieties, e.g. Fulton's book. (The characteristic zero assumption is irrelevant.) $\endgroup$
    – Piotr Achinger
    Commented Nov 5, 2019 at 11:01

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