\[
\dynkin[Coxeter]{F}{4} = \dynkin[label, arrows=false, edge length=.75cm,o/.style=black]F4
\]


> Conventions on Coxeter diagrams: 
> https://file.notion.so/f/s/962e72a3-fa3a-45ec-84a3-2bb0e7699a7b/Alexeev-Nikulin_Del_Pezzo_and_K3_Surfaces.pdf?id=7e4996b4-08e6-4bd2-ac49-8d23080f7577&table=block&spaceId=7cb2f7c7-7373-4d11-91ab-284625335dc8&expirationTimestamp=1683905929597&signature=fMZC5iiiI0MbbvXkvnI4JyGwzN6JjCSDIU7Z6itmcUE&downloadName=AN06.pdf#page=64&zoom=auto,-155,146

The interpretation of these Coxeter diagrams in terms of root systems:

Needs some notation from \cite{AN06}:

\[
\begin{array}{ll}
V(M) &\text {the light cone } V(M)=\left\{x \in M \otimes \mathbb{R} \mid x^2>0\right\} \text { of a hyperbolic lattice } M \\
V^{+}(X) &\text {the half containing polarization of the light cone } V\left(S_X\right) \\
\mathcal{L}(S)=V^{+}\left(S_X\right) / \mathbb{R}^{+} & \text {the hyperbolic space of a surface } S \\
W^{(2)}(M) & \text{the group generated by reflections in all $f \in M$ with $f^2=-2$} \\
W^{(4)}(M) & \text{the group generated by reflections in all $(-4)$ roots of $M$} \\
W^{(2,4)}(M) & \text{the group generated by reflections in all $(-2)$ and $(-4)$ roots of $M$} \\
\mathcal{M}^{(2)} & \text {a fundamental chamber of } W^{(2)}(S) \text { in } \mathcal{L}(S) \\
\mathcal{M}^{(2,4)} & \text {a fundamental chamber of } W^{(2,4)}(S) \text { in } \mathcal{L}(S) \\
P^{(2)}\left(\mathcal{M}^{(2,4)}\right) & \text {all }(-2) \text {-roots orthogonal to } \mathcal{M}^{(2,4)}  \\
P^{(4)}\left(\mathcal{M}^{(2,4)}\right) & \text {all }(-4) \text {-roots orthogonal to } \mathcal{M}^{(2,4)}  \\
(X, \theta) &\text{a $\Kthree$ with involution $\theta$} \\
X^\theta &\text{the fixed locus of an involution} \\
P(X)_{+I} & \text {the subset of exceptional classes of }(X, \theta)  \text{ of type } \rm{I}
\end{array}
\]

## Edge notation

- Vertices corresponding to different elements $f_1, f_2 \in P\left(\mathcal{M}^{(2,4)}\right)$ are not connected by any edge if $f_1 \cdot f_2=0$. 

- Simple edges of weight $m$ (equivalently, by $m-2$ simple edges if $m>2$ is small):
\[
\tikz{
    \node [style=white node] (0) at (0, 0) {};
    \node [style=white node] (1) at (3, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f_1$};
    \node [style=none] (5) at (3, 0.4) {$f_2$};
    \draw (0) to (1);
    \node [style=none] at (1.5, 0.4) {$m$};
}
\qquad \implies \qquad
\frac{2 f_1 \cdot f_2}{\sqrt{f_1^2 f_2^2}}=2 \cos \frac{\pi}{m}, \quad m \in \mathbb{N}
    \]
    
- Thick edges:
\[
\tikz{
    \node [style=white node] (0) at (0, 0) {};
    \node [style=white node] (1) at (3, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f_1$};
    \node [style=none] (5) at (3, 0.4) {$f_2$};
    \draw [line width=2pt] (0) to (1);
}
\qquad \implies \qquad
\frac{2 f_1 \cdot f_2}{\sqrt{f_1^2 f_2^2}}=2
    \]
    
- Broken edges of weight $t$:
\[
\tikz{
    \node [style=white node] (0) at (0, 0) {};
    \node [style=white node] (1) at (3, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f_1$};
    \node [style=none] (5) at (3, 0.4) {$f_2$};
    \node [style=none] at (1.5, 0.4) {$t$};
    \draw [dashed] (0) to (1);
} 
\qquad \implies \qquad
\frac{2 f_1 \cdot f_2}{\sqrt{f_1^2 f_2^2}}=t>2
    \]
    
- A vertex corresponding to $f \in P^{(4)}\left(\mathcal{M}^{(2,4)}\right)$ is black:
\[
\tikz{
    \node [style=black node] (0) at (0, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f$};
} 
\qquad \implies \qquad
f^2 = -4?
    \]
    
- It is white if $f \in P^{(2)}\left(\mathcal{M}^{(2,4)}\right)$:
\[
\tikz{
    \node [style=white node] (0) at (0, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f$};
} 
\qquad \implies \qquad
f^2 = -2?
    \]
    
- It is double-circled white if $f \in P(X)_{+I}$ (i.e. it corresponds to the class of a rational component of $\left.X^\theta\right)$.
\[
\tikz{
    \node [style=white node, double] at (0, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f$};
} 
\qquad \implies \qquad
??
    \]

Interpreting this geometrically: consider the cycle of $2\bar{k}$ white vertices cycling between plain and double-circled:

> Second to last paragraph here: https://arxiv.org/pdf/2208.10383.pdf#page=41&zoom=140,99,262

- See \cite{AE22}

- Each edge on the outer cycle corresponds to $\mathbb{P}^2$

- Single circle vertices (with odd $i$) corresponds to a line in $\mathbb{P}^2$

- Double-circled vertices (with even $i$) correspond to conics on the $\mathbb{P}^2$

- Explicit example worked out in \cite[\S 5]{AET23}. 

> % https://arxiv.org/pdf/1903.09742.pdf#page=28&zoom=170,-70,442

- It seems like that from the Coxeter diagram, you draw the fan of a toric surface, you compute the charge, and then, if this is not 24, you fix it by blowing up some non-torus fixed points along the toric boundary.