Exercise 111.59.3 (0DT2)—The Stacks project

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Part 9: Miscellany
Chapter 111: Exercises
Section 111.59: Schemes, Final Exam, Spring 2017
Exercise 111.59.3 (cite)

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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:

What is the genus of $X$ ?
Give an upper bound for the gonality ¹ of $X$ .

[1] The gonality is the smallest degree of a nonconstant morphism from

$X$ to

$\mathbf{P}^1_ k$ .

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