The Stacks project

Exercise 111.59.3. Let $k$ be an algebraically closed field. Let $\ell > 100$ be a prime number different from the characteristic of $k$. Let $X$ be the nonsingular projective model of the affine curve given by the equation

\[ y^\ell = x(x - 1)^3 \]

in $\mathbf{A}^2_ k$. Answer the following questions:

  1. What is the genus of $X$?

  2. Give an upper bound for the gonality1 of $X$.

[1] The gonality is the smallest degree of a nonconstant morphism from $X$ to $\mathbf{P}^1_ k$.

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View Exercise 111.59.3 as pdf