Noncommutative Smooth Models

We determine the central simple algebras D over a functionfield K of trancendence degree two which admit a model of smooth Cayley-Hamilton algebras. This happens if and only if there is a smooth model S of K such that the ramification divisor of a maximal S-order in D is a disjoint union of smooth curves. Further, we prove that the Brauer-Severi fibration of smooth models which are in addition maximal orders is a flat morphism and determine the number of irreducible components of the fibers.