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

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 $0$.

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