# Intersection of (Toric varieties) and (Flag varieties)

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.

• 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. Commented Dec 2, 2022 at 7:48
• 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.
– Dave
Commented Dec 2, 2022 at 22:17

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.