Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $X$ be Klein's quartic curve given by $x^3y + y^3z+z^3x=0$ in $\mathbf{P}^2$. It is isomorphic to $X(7)$.

How do I easily show that $X$ is not hyperelliptic?

I can see that $X$ is of genus $3$ and has gonality $\leq 3$ (consider the projection). I'm trying to prove that it has gonality $3$.

More generally, what is a computationally feasible way to check if a curve is not hyperelliptic?

Note that I'm not really asking for a criterion. For example, to check if a variety is normal you could try to show that it is regular (which is easier to me).

Is the obvious morphism $X\to \mathbf{P}^1$ of degree $3$ Galois? That is, do we have that $X$ is a cyclic cover of degree $3$?

share|cite|improve this question
up vote 9 down vote accepted

For a smooth curve $X$ of genus $g\geq 3$ (like the Klein quartic, which has genus $g=3$ as you remarked), the criterion you want is (Miranda , Chap.VII, Prop. 2.1):

$X$ is not hyperelliptic $\iff$ the canonical map $X\to \mathbb P^{g-1}$ is an embedding.

Conclude by remembering that if $X\subset \mathbb P^{g-1} $ is already embedded as a curve of of degree $2g-2$ (but not included in a hyperplane), then the canonical map is an embedding: Griffiths-Harris, page 247.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.