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For me, a variety $X$ is assumed to be irreducible and normal over some algebraically closed field $k$ of characteristic zero.

I call $X$ a toric variety if it has an algebraic torus $T$ which embeds $T\hookrightarrow X$ as an open subset such that the action of $T$ on itself (multiplication) extends to an algebraic action on all of $X$.

I call $X$ a flag variety if $X\cong G/P$ for some connected reductive linear algebraic group $G$ and some parabolic subgroup $P\subseteq G$. Note that flag varieties are smooth and projective.

My question is this:

Which varieties $X$ are simultaneously a toric variety and a flag variety?

Certainly such varieties $X$ need to be smooth and projective. Some examples:

(1) If $X=\mathbb{P}^n$, then this is certainly a toric variety, and it is a flag variety via $SL_n/P_n$ where $P_n$ is a maximal parabolic subgroup (not equal to all of $SL_n$).

(2) Products of the above example, e.g. $\mathbb{P}^1\times\mathbb{P}^1$, since toric and flag varieties are closed under taking products.

(3) Points since a point is clearly toric (the trivial torus action) and is of the form $G/G$.

(4) As a non-example, the homogeneous space $SL_3/B$ is a flag variety but not a toric variety ($B$ is the Borel subgroup). Indeed, it is clearly a flag variety, but its Cox ring is not a polynomial ring, so it cannot be toric (see Corollary 2.10 in "Mori dream spaces and GIT" by Hu and Keel; and see Example 4.1 in "The Cox ring of a spherical embedding" by Gagliardi for the Cox ring calculation).

I'm not sure if there is a "nice" general answer to my question. Any help is appreciated.

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  • $\begingroup$ The Grassmannians, even $G(2,4)$, are not toric, since they are given by trinomials like $p_{1,2}p_{0,3}-p_{0,2}p_{1,3}+p_{0,1}p_{2,3}=0$ (plucker) and not binomials. $\endgroup$ Dec 2, 2022 at 7:48
  • $\begingroup$ Thanks for the example. I think flag varieties with "small" parabolics will give this kind of behaviour in general (in your example, the parabolic is not maximal). My thought is that maybe the criteria is $G/P$ with $P$ a maximal parabolic, but I'm not sure why other parabolics won't work in general. $\endgroup$
    – Dave
    Dec 2, 2022 at 22:17

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Partial flag varieties satisfy a property called "being $\mathscr D$-affine". (This is the Beilinson-Bernstein localisation theorem). The following article "$\mathscr D$-affinity and toric varieties" prove that the only projective toric varieties that are $\mathscr D$-affine are product of projective spaces.

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  • $\begingroup$ Thanks for the reference, this is excellent! So the examples (2) that I list above are the only ones, which is a pretty nice answer to my question. $\endgroup$
    – Dave
    Dec 3, 2022 at 22:29

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