I just got through reading section I.6 in Hartshorne, and I have no idea why anybody would ever have invented these concepts. I agree that what you can do with it is pretty neat, but how would anyone have ever thought of it?
In case anyone needs the definitions, here they are.
If $K$ is a function field of dimension $1$ and $k$ is a subfield of $K$ such that the valuation function $v$ satisfies $v(x) = 0$ for all $ x \in k- \{0\}$, then $v$ is a valuation of $K/k$ and $R := \{ x \in K| v(x) \geq 0 \}$ is a valuation ring of $K/k$. Given a particular $K$, define $C_K$ to be the set of all discrete valuation rings of $K/k$. Topologize $C_K$ by declaring the closed sets to be the finite subsets, and $C_K$ itself. Now an abstract non-singular curve is an open subset.
These things do have a handful of really nice theorems Hartshorne proves in the section, but I would never have thought to look at such objects. In fact it feels completely unmotivated.
So my question is either a) how would you think this up, or b) can we prove the main theorem of this section, which is that every curve is birational to a non-singular projective curve, without all of this?