TITLE: Klein's curve (algebraic geometry)
QUESTION [9 upvotes]: 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!

REPLY [11 votes]: 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.)