TITLE: Effectiveness of the distinguished theta characteristic in characteristic 2
QUESTION [9 upvotes]: Let $k$ be an algebraically closed field of characteristic 2. Let $C$ be a (smooth projective connected) curve over $k$. Can there exist a rational function on $C$ whose differential is holomorphic but nonzero? (Note that every perfect square has zero differential.)
Alternate formulation: at the end of his paper "Theta characteristics of an algebraic curve", Mumford points out that the canonical sheaf on $C$ has a distinguished square root $\mathcal{L}$: for any rational function $f$ on $C$ which is not a perfect square, the divisor $(df)$ is even and the class of $\frac{1}{2}(df)$ does not depend on $f$. The question is then whether $\mathcal{L}$ can admit a nonzero section.
For example, it is an entertaining exercise to check (from a Weierstrass model) that this can never occur for ordinary elliptic curves.

REPLY [3 votes]: Let $F:X\to X$ be the relative Frobenius morphism. It is a finite morphism of degree $2$, and we have an exact sequence 
$$0\to \mathcal{O}_X\to F_*\mathcal{O}_X \to L \to 0$$
for some invertible sheaf $L$ on $X$. The standard formula for the canonical class of a finite morphism tells us that 
$\omega_X = F^*(\omega_X\otimes L^{-1}) = \omega_X^{\otimes 2}\otimes 
L^{\otimes -2}.$
This implies that $L = \theta$ is a theta characteristic on $X$. I believe this is the theta characteristic discussed by Mumford. Now apply the cohomology to the first exact sequence to get that $H^1(X,\theta) \cong H^0(X,\theta)^*$ is the cokernel of the Frobenius map $H^1(X,\mathcal{O}_X) \to H^1(X,\mathcal{O}_X)$.
So, $\theta$ is effective if and only if the curve is not ordinary.