Klein's curve (algebraic geometry)

Question

I can't find any information about the canonical ring of Klein's quartic curve (the one with 168 automorphisms). I would imagine there is a lot known about the structure of this ring. Can anybody help me please? Thank you!

Answer 1

Klein quartic, $X$, is a smooth degree $4$ plane curve, given by the equation $F(x,y,z) = x^3 y + y^3 z + z^3 x = 0$. Doesn't this mean that the canonical ring is $k[x,y,z]/(x^3 y + y^3 z + z^3 x)$?

Details: This embedding is the canonical embedding, since we compute by adjunction that $\mathcal{O}(1)$ is the canonical bundle. Thus, we have an injection $k[x,y,z]/F \hookrightarrow \bigoplus_{n=0}^{\infty} H^0(X, K^{\otimes n})$. The Hilbert series of the left hand side is $(1-t^4)/(1-t)^3 = 1+3t + \sum_{n \geq 2} (4n-2) t^n$. The Hilbert series of the canonical ring is $1+gt + \sum_{n \geq 2} ((2g-2)n-(g-1)) t^n$. These match, so the map is an isomorphism. (Actually, I think I recall that the canonical ring of a nonhyperelliptic curve is always generated in degree $1$, but it seemed easier to check the Hilbert series in this case.)

Answer 2

There are many resources for this, one of easy treatments can be found in

https://www.math.hmc.edu/~ursula/teaching/math189/finalpapers/julia.pdf

Answer 3

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 and in general the book The Eightfold Way is online. In the translation of Klein's original work we find this futuristic image:

Also in German. 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 (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