There are a few properties from Hartshorne IV on curves that I am trying to verify. Let $D$ be an effective divisor on a curve (integral scheme of dimension 1, proper over $k$, regular) $X$, $\dim |D| = 1$, and $f$ the induced morphism to $\mathbb{P}^1$.

- The degree of $D$ is the degree of $f$, where $\deg(f) = [K(X) : K(\mathbb{P}^1)]$.
- In lemma IV.4.2, Hartshorne seems to claim that all points in the support of $D$ are in the same fiber.
- Again in lemma IV.4.2, Hartshorne seems to claim that if $[K(X) : K(\mathbb{P}^1)]$ is Galois, then an automorphism of the Galois group permutes elements of the fiber.