When are these definitions of “toric variety” equivalent?

Let $k$ be an algebraically closed field. Let $X$ be an integral $k$-scheme, separated and of finite type over $k$. Let $d := \dim X$, let $T := (\mathbb{G}_{m,k})^{d}$ be the $d$-dimensional torus, and suppose we have an action of $T$ on $X$ over $k$ given by the action morphism $\sigma : T \times_{k} X \to X$. Let $$\mathrm{stab}_{\sigma} : Z_{\sigma} \to X$$ be the stabilizer of $\sigma$, namely the pullback of the morphism $(\sigma,p_{2}) : T \times_{k} X \to X \times_{k} X$ via the diagonal $\Delta_{X/k} : X \to X \times_{k} X$. For a $k$-point $x \in X(k)$, let $$\alpha_{x} : T \to X$$ denote the orbit morphism of $x$, namely the composition $\mathrm{id}_{T} \times x : T \to T \times_{k} X$ with $\sigma$, and we denote by $T \cdot x \subset X$ the (set-theoretic) image of $\alpha_{x}$. By the closed orbit lemma, we have that $T \cdot x$ is a locally closed subset of $X$.

Depending on the source, the scheme $X$ is called a toric variety if it satisfies one of the following conditions (ordered roughly from strongest to weakest):

1. The scheme $X$ is given by a construction involving fans (e.g. [4, Theorem 1.4] or [1, Section 2.1]).

2. There is a $T$-equivariant open immersion $j : T \to X$, where $T$ acts on itself by left multiplication.

3. The action $\sigma$ has generically trivial stabilizers, i.e. there exists a dense open subscheme $U \subseteq X$ such that the restriction $\mathrm{stab}_{\sigma}^{-1}(U) \to U$ is an isomorphism.

4. The action $\sigma$ has generically finite stabilizers, i.e. there exists a dense open subscheme $U \subseteq X$ such that the restriction $\mathrm{stab}_{\sigma}^{-1}(U) \to U$ is a quasi-finite morphism.

5. There is a $k$-point $x \in X(k)$ such that $T \cdot x$ is an open subset of $X$.

6. The action $\sigma$ is faithful, i.e. the corresponding morphism of group sheaves $T \to \underline{\mathrm{Aut}}_{(\mathrm{Sch}/k)}(X)$ on the category of $k$-schemes $(\mathrm{Sch}/k)$ is injective.

Do the above conditions imply any others (possibly with mild assumptions about $X$)?

At the moment, I am most interested in whether (4) implies (2).

Here is what I have so far: In (1), the scheme $X$ is obtained by equivariantly gluing certain affine $T$-schemes which satisfy (2), hence (1) implies (2). If $X$ is normal, then (2) implies (1) by Sumihiro's theorem [4, Theorem 1.5]. For (2) implies (3), take any $k$-point $x \in X(k)$ lying in the image of $j$. Clearly (3) implies (4). For the equivalence of (4) and (5), use the orbit-stabilizer theorem for algebraic groups (i.e. that $\dim (T \cdot x) + \dim (x^{-1}(\mathrm{stab}_{\sigma})) = \dim T$, see [2, Proposition 3.20]) and the assumption that $T$ and $X$ have the same dimension (since $T \cdot x$ is locally closed, saying that $T \cdot x$ is open is the same as saying that $\dim (T \cdot x) = d$). For (2) implies (5), either compose previous implications or take $U = j(T)$. For (2) implies (6), restriction via $j$ gives a map $\underline{\mathrm{Aut}}_{(\mathrm{Sch}/k)}(X) \to \underline{\mathrm{Aut}}_{(\mathrm{Sch}/k)}(T)$ such that the composition $T \to \underline{\mathrm{Aut}}_{(\mathrm{Sch}/k)}(X) \to \underline{\mathrm{Aut}}_{(\mathrm{Sch}/k)}(T)$ is injective, hence the first arrow is injective.

References:

[1] Elizondo, Lima-Filho, Sottile, Teitler, "Arithmetic Toric Varieties", Mathematische Nachrichten, vol. 287, no. 2-3 (2014) link

[2] Hoskins, "Moduli problems and geometric invariant theory", online course notes (2015)

[3] Kempf, Knudsen, Mumford, Saint-Donat, "Toroidal Embeddings 1", Springer Lecture Notes in Mathematics, vol. 339 (1973)

[4] Oda, "Convex Bodies and Algebraic Geometry", Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3 (1985)

• Usually I think conditions (1) or (2) are what is meant by a toric variety, with the difference being whether one requires normality or not (as you note). Anyway, condition (4) does not imply (2), as you can see from $T=k^*=X$ acting on itself by $t\cdot z = t^2z$. – Dave Anderson Nov 2 '17 at 1:52
• @DaveAnderson Thanks. Do you know of an example of $(X,\sigma)$ satisfying (4) such that $X$ does not admit a different action $\sigma' : T \times_{k} X \to X$ so that (2) holds with respect to $(X,\sigma')$? – Minseon Shin Nov 2 '17 at 3:27

As noticed by Dave Anderson, (4) does not imply (2) because the generic stabilizer might be finite but non-trivial. But still, let $E$ be the stabilzer subgroup scheme of $x$ as in (5). Because $T$ is abelian, $E$ is normal in $T$. Hence $E$ acts trivially on the open set $Tx$. The fixed point scheme of $E$ being closed this implies that $E$ acts trivially on all of $X$. Hence the action of $T$ on $X$ factors through the torus $T_0=T/E$ for which then (2) is satisfied.
To see the claim one can e.g. use that subgroup schemes of a torus are rigid. More precisely, let $U\subseteq X$ be open, dense over which $Z_\sigma$ is flat. Then one can show that it is a constant subgroup scheme of $T\times U$. Therefore any fiber $E$ acts trivially on $U$, hence on $X$. Thus $E=1$ by (6).