Intereting Posts

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What does area represent?
Express $1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb + \frac{1}{n + 1}\binom{n}{n}$ in a simplifed form
Is there a $k$ for which $k\cdot n\ln n$ takes only prime values?
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Expected Values of Operators in Quantum Mechanics
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Amazing integrals and how is solved it
Banach Spaces – How can $B,B',B'', B''', B'''',B''''',\ldots$ behave?
KKT: Explain visually the optimality condition $F_0\cap G_0\cap H_0=\emptyset$
What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 – z^3$

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 $3$.

- Rational map on smooth projective curve
- Nonsingular projective variety of degree $d$
- equation of a curve given 3 points and additional (constant) requirements
- What is the best way to see that the dimension of the moduli space of curves of genus $g>1$ is $3g-3$?
- Divisor on curve of genus $2$
- Conic by three points and two tangent lines

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 $3$?

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- Pullback and Pushforward Isomorphism of Sheaves

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.

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