The Stacks project

Exercise 111.46.2. Let $k$ be an algebraically closed field. Let $X$ be a smooth curve over $k$. Let $r \geq 1$. Show that the closed subset

\[ D \subset X \times X^ r = X^{r + 1} \]

whose closed points are the tuples $(x, x_1, \ldots , x_ r)$ with $x = x_ i$ for some $i$, has an invertible ideal sheaf. In other words, show that $D$ is an effective Cartier divisor. Hints: reduce to $r = 1$ and use that $X$ is a smooth curves to say something about the diagonal (look in books for this).

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View Exercise 111.46.2 as pdf