Can one characterise a global field geometrically?

Question

A global field is either a

1. a finite extension of the rationals

2. a finite extensions of $F_q(t)$

Alternatively, the second is the function field of an algebraic curve over a finite field.

Is there a geometric equivalent to first that is parallel to this?

Answer 1

It is the function field of an integral scheme of finite type and integral over $\mathbb{Z}$.