Example 53.13.6. This example will show that the genus can change under a purely inseparable morphism of nonsingular projective curves. Let $k$ be a field of characteristic $3$. Assume there exists an element $a \in k$ which is not a $3$rd power. For example $k = \mathbf{F}_3(a)$ would work. Let $X$ be the plane curve with homogeneous equation

$F = T_1^2T_0 - T_2^3 + aT_0^3$

as in Section 53.9. On the affine piece $D_+(T_0)$ using coordinates $x = T_1/T_0$ and $y = T_2/T_0$ we obtain $x^2 - y^3 + a = 0$ which defines a nonsingular affine curve. Moreover, the point at infinity $(0 : 1: 0)$ is a smooth point. Hence $X$ is a nonsingular projective curve of genus $1$ (Lemma 53.9.3). On the other hand, consider the morphism $f : X \to \mathbf{P}^1_ k$ which on $D_+(T_0)$ sends $(x, y)$ to $x \in \mathbf{A}^1_ k \subset \mathbf{P}^1_ k$. Then $f$ is a morphism of proper nonsingular curves over $k$ inducing an inseparable function field extension of degree $p = 3$ but the genus of $X$ is $1$ and the genus of $\mathbf{P}^1_ k$ is $0$.

Comments (2)

Comment #7482 by Hao Peng on

Shouldn't the map to P^1 be projection to x coordinates instead of y cooediantes?

There are also:

• 7 comment(s) on Section 53.13: Inseparable maps

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CD1. Beware of the difference between the letter 'O' and the digit '0'.