Relation between fan and toric variety
By Sarah Smith
I understand what a fan is, but I don't really know how it associated to a toric variety. Could a non-toric variety has a fan?
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$\begingroup$Note: I'm not an expert on the subject, I'm someone who's had a similar question in the last few days and has been trying to find an answer. I can't guarantee that everything here is 100% correct, but I think this is basically the right idea.
Suppose you have a variety $X$ with a torus action $T \curvearrowright X$. The cocharacter lattice $N := \operatorname{Hom}(\mathbb{G}_m, T)$ is the set of one parameter subgroups $\lambda : \mathbb{G}_m \to T$ in $T$.$^{[1]}$ The elements of $N$ are $$\lambda_{\overline{a}}(t) := (t^{a_1}, t^{a_2}, \ldots, t^{a_n})$$ where $\overline{a} \in \mathbb{Z}^n$. In particular, we have $N \cong \mathbb{Z}^n$, so it is indeed a lattice.
Suppose we pick some $\lambda \in N$. If we're working over the complex numbers, then it makes sense to talk about $$\lim_{t \to 0} \lambda(t)$$ which may or may not exist. Further, if we have two different one-parameter subgroups $\lambda_1, \lambda_2 : N$, we can ask if $$\lim_{t \to 0} \lambda_1(t) = \lim_{t \to 0} \lambda_2(t)$$
One way to describe the fan $\Sigma$ of $T \curvearrowright X$ is to say that, for each point $x$, you associate the cone $$C_x := \{ \lambda \in N \; | \; \lim_{t \to 0} \lambda(t) = x \}$$
As it turns out each $C_x$ will actually be a cone, and only finitely many of them will be nonempty. You then have $\Sigma := \{ C_x \; | \; x \in X\}$.
If $X$ is a normal variety, you can reconstruct $X$ (together with the $T$ action) as $X_\Sigma$. If it is not a normal variety, then I believe $X_\Sigma$ will be the normalization of $X$.
Let's try it out. Consider $T^2 \curvearrowright \mathbb{A}^2$ by $(s, t) \cdot (x, y) = (s x, t y)$.$^{[2]}$ Then the one-parameter subgroups are given by $\lambda_{a, b}(t) = (t^a, t^b)$. We have
$$\lim_{t \to 0} \lambda_{a,b}(t) = \lim_{t \to 0} (t^a, t^b) = \begin{cases} (0,0) & \text{ if } a \geq 0, b \geq 0 \\ \text{undefined} & \text{ otherwise } \end{cases}$$
So $$C_x = \begin{cases} \langle f_1, f_2 \rangle & \text{ if } x = (0,0) \\ \emptyset & \text{ otherwise } \end{cases}$$
Consequently the fan consists of a single cone which is the first quadrant of $N$, as desired.
Regarding non-toric varieties: you at least need a torus action to carry out the construction above. You could try and see what happens if you take something with a torus action that's not a toric variety.
[1] If you're not familiar with the algebraic geometry notation, $\mathbb{G}_m$ is the multiplicative group, also known as $\operatorname{GL}_1$. Working over the complex numbers, it's just the group of nonzero complex numbers under multiplication, sometimes denoted $\mathbb{C}^\ast$.
[2] $\mathbb{A}^2$ is two-dimensional affine space, or as a complex manifold $\mathbb{C}^2$.
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