# Wednesday, September 06

:::{.remark}
$\S 3.2$: The Orbit-Cone correspondence.
There is a 1-to-1 correspondences between cones of \( \Sigma \) and orbits of $T\actson X_ \Sigma$.
Here $x,y$ are in the same torus orbit if $\exists t\in T$ with $t.x = y$.
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:::{.example title="?"}
For $X_\Sigma = \PP^1$ with coordinates $[x: y]$, the action is $t.[x:y] = [x: ty]$. 
This has fixed points $[1: 0]$ and $[0: 1]$, and there is an orbit $\ts{[x: y] \st x,y\neq 0} \cong \cstar$.
There are three cones in the fan for $\PP^1$, namely $\sigma_+ \da \RR_{> 0}, \sigma_- \da \RR_{ < 0}, \sigma_0 \da \ts{0}$.
Here $\sigma_+$ corresponds to $[1: 0]$, $\sigma_-$ to $[0: 1]$, and $\sigma_0$ to $\cstar$.
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:::{.remark}
For any $u\in N \intersect \sigma^\circ$ a lattice point in the interior of a cone, recall $u$ corresponds to a cocharacter.
This defines a map $\lambda^u: \cstar \to (\cstar)^n$ where $t\mapsto (t^{u_1},\cdots, t^{u_n})$.
Since $(\cstar)^n \embeds X_\Sigma$, the composition embeds $\cstar \embeds X_\Sigma$.
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:::{.theorem title="?"}
For any point $u\in N$, the limit $t\to 0$ exists in $X_\Sigma$.
This point $\lim_{t\to 0} (t^{u_1},\cdots, t^{u_n})$ is in some orbit $\OO_\sigma$, which we declare to be the orbit corresponding to $\sigma$.
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:::{.theorem title="?"}
Any $u\in N$ in the relative interior of $\sigma$ yields the same orbit $\OO_{ \sigma}$.
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:::{.example title="?"}
For $\AA^1$, take $\sigma = \RR_{ > 0} \subseteq N_\RR = \RR$. Consider the composition $\cstar_t \embeds \cstar \embeds \AA^1$ where $t\mapsto t^n$ for $u = (n)\in N$.
Check that $t^n \convergesto{t\to 0} 0$ if $n > 0$, and for $\sigma' = \ts{0}$ we have $t^n \to 1$.
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:::{.example title="?"}
For $\PP^1$, it depends on $n > 0, n=0, n < 0$, and correspondingly $t^n \to 0, 1,\infty$ respectively, corresponding to $[1: 0], \cstar, [0: 1]$.
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