Not sure what "canonical" means in this context. Half of algebraic geometry is called "canonical ring" or "fundamental class" or some other vacuous term.
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Please consider Elkies [The Klein Quartic in Number Theory](http://library.msri.org/books/Book35/files/elkies.pdf) and in general the book **The Eightfold Way** is online. In the translation of [Klein](http://library.msri.org/books/Book35/files/klein.pdf)'s original work we find this futuristic image:
[![enter image description here][1]][1]
Also in [German](https://eudml.org/journal/10142). The translated version has improved images
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I have no idea what the above really means, but it sounds like we get really lucky. These results are due to Riemann -- long before Serre, Grothendieck, Hartshorne or Shafarevich. How could he have anything more complex in mind?
The only ring mention in Klein's paper is $\mathbb{C}[x,y,z]/(x^3y + y^3z+z^3x)$ -- the one David Speyer wrote -- as well as $SL(2, \mathbb{F}_7)$ and $\mathbb{Z}(\frac{1 + \sqrt{-7}}{2})$. Which one is the canonical ring?
Since $x, y, z$ are the coordinates of an equation $x^3 y + y^3 z + z^3 x = 0$, then $x, y, z$ can be thought of as parameterizing solutions to the set $$\{x^3 y + y^3 z + z^3 x = 0\}$$
So the coordinate ring might be the coordinate ring of $\mathbb{P}^N$ mod a single relation:
$$ \mathbb{C}[x,y,z]/(x^3 y + y^3 z + z^3 x ) $$
In fact, $x, y, z$ are **sections** of the line bundle $\mathcal{O}_X$ and if $i+j+k = n$ then maybe we can multiply them to get sections of the other sheafs:
$$ x^i y^j z^k \in H^0(X, n \mathcal{O}_C)$$
Looks like the coordinate ring is the direct sum of all of these possibilities excepting for the possibility that $x^3 y + y^3z + z^3 x = 0$.
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If we then re-read Klein, there is even more to say:
\begin{eqnarray}
x &=& \sum_{\beta \equiv 1 \mod \sqrt{-7}} \mathrm{Re}(\beta) \, q^{\beta \overline{\beta}/7} \\
y &=& \sum_{\beta \equiv 2 \mod \sqrt{-7}} \mathrm{Re}(\beta) \, q^{\beta \overline{\beta}/7} \\
z &=& \sum_{\beta \equiv 4 \mod \sqrt{-7}} \mathrm{Re}(\beta) \, q^{\beta \overline{\beta}/7}
\end{eqnarray}
and these theta functions explicitly solve the equation $x^3 y + y^3 z + z^3 x = 0$ in terms of modular functions over $\mathbb{H}/SL(2, \mathbb{F}_7)= \langle z \mapsto z + 7, z \mapsto - \frac{1}{z} \rangle $. These sections $x,y,z \in \mathbb{H}$ are **modular cusp forms of weight 2** over $X(7)$ -- one must verify that $x, y, z, \neq 0$ exceept at the "cusps".
The fact that these theta functions parameterize a curve both on $\mathbb{H}$ and in projective space $\mathbb{P}^3$ does not seem trivial at all. A lot of effort may go into showing the rather strange-looking coordinates we chose do not have exceptional behavior.
And these are generators of your canonical ring!
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[Birational Geometry Old and New](http://www.ams.org/journals/bull/2009-46-01/S0273-0979-08-01233-0/home.html) (Antonella Grassi) has this much to say:
> **Example-Theorem 29**: If $C$ is a curve of genus $g \geq 2$ the divisor $3 K_C$ is *very ample*. There is an embedding $\phi: C \to \mathbb{P}^N$ such that if $H$ is a hyperplane $H \cdot C = 3K_C$. Then the canonical ring $R(C, K_C)$ can be reconstructed from the coordinate ring of $\mathbb{P}^N$. In particular, $R$ is finitely generated. In fact $3K_C$ determines a *pluricanonical embedding*.
Antonella Grassi explain that nothing too exceptional occurs, so that for example $K_C = \mathcal{O}(-1)$, and the coordinates $x,y,z$ can be identified with hyperplanes $\{ x=0\}, \{ y = 0\}, \{z=0\}$ and then $[x],[y],[z]$ are **hyperplane divisors** and $H\cdot C$ is just setting $x$, $y$, or $z = 0$.
For much more please turn to the standard texts:
* Hartshorne **Algebraic Geometry**
* Shafarevich **Basic Algebraic Geometry**
[1]: https://i.sstatic.net/DRigf.png