The Stacks project

Exercise 109.63.5. Let $k$ be an algebraically closed field of characteristic $0$. Let

\[ X : T_0^ d + T_1^ d - T_2^ d = 0 \subset \mathbf{P}^2_ k \]

be the Fermat curve of degree $d \geq 3$. Consider the closed points $p = [1 : 0 : 1]$ and $q = [0 : 1 : 1]$ on $X$. Set $D = [p] - [q]$.

  1. Show that $D$ is nontrivial in the Weil divisor class group.

  2. Show that $d D$ is trivial in the Weil divisor class group. (Hint: try to show that both $d[p]$ and $d[q]$ are the intersection of $X$ with a line in the plane.)

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