This question is based on a line in the proof of corollary IV.3.3 in Hartshorne's Algebraic Geometry.
The first line of the proof goes: "if $D$ is an ample divisor (on a curve $X$), then some multiple is very ample, so $nD\sim H$, where $H$ is a hyperplane section for a projective embedding."
This line suggests that there is some obvious relationship between hyperplane sections and very ample divisors, namely that any very ample divisor is linearly equivalent to a hyperplane section.
Why is this true? Is this done somewhere else in Hartshorne? (I've been looking for a long time, but I can't seem to find it anywhere else!)