Algebraic function field of one variable as a scheme

Question

Let $F|K$ be an algebraic function field of one variable. How can it be considered as a scheme? And how can one define geometric and arithmetic genus for it?!

Answer 1

There exists $t\in F$ transcendental over $K$ such that $F/K(t)$ is finite. Let $A$ (resp. $B$) be the integral closure of $K[t]$ (resp. $K[1/t]$) in $F$. Note that $B_{1/t}$ and $A_t$ are equal, so we can glue the spectra of $A$ and $B$ along $D(t) = D(1/t)$, where the LHS is an open subset of the spectrum of $A$ and the latter is an open subset of the spectrum of $B$.

The scheme $X$ which is obtained is naturally an integral $K$-scheme with function field $F$ and even has a natural morphism to $\mathbb{P}^1$ corresponding to the rational function $t$. Usual scheme theory applies to define a genus.

When $F/K(t)$ is separable, $A$ and $B$ are Dedekind rings and finitely generated modules over $K[t]$ and $K[1/t]$, which entails that $X$ is a noetherian Dedekind scheme hence is regular. So you may easily enough define a notion of arithmetic and geometric genus.