The Stacks project

Proof. Recall that we have a universal $k$-derivation $d : \mathcal{O}_ X \to \Omega _{X/k}$ and hence $D = \theta \circ d$ for some $\mathcal{O}_ X$-linear map $\theta : \Omega _{X/k} \to \mathcal{O}_ X$. Recall that $\Omega _{X/k} \cong \omega _ X$, see Lemma 53.4.1. By Riemann-Roch we have $\deg (\omega _ X) = 2g - 2$ (Lemma 53.5.2). Thus we see that $\theta$ is forced to be zero if $g > 1$ by Varieties, Lemma 33.44.12. This proves part (1). If $g = 1$, then a nonzero $\theta$ does not vanish anywhere and if $g = 0$, then a nonzero $\theta$ vanishes in a divisor of degree $2$. Thus parts (2) and (3) follow if we show that vanishing of $\theta$ at a closed point $x \in X$ is equivalent to the statement that $D$ fixes $x$ (as defined above). Let $z \in \mathcal{O}_{X, x}$ be a uniformizer. Then $dz$ is a basis element for $\Omega _{X, x}$, see Lemma 53.12.3. Since $D(z) = \theta (dz)$ we conclude. $\square$