Noncommutative Smooth Models

TL;DR

This paper characterizes when certain central simple algebras over function fields of transcendence degree two admit smooth models, linking algebraic properties to geometric conditions on ramification divisors and analyzing the structure of associated Brauer-Severi fibrations.

Contribution

It provides a complete characterization of smooth Cayley-Hamilton algebra models over function fields of two variables, connecting algebraic and geometric conditions, and analyzes the structure of their Brauer-Severi fibrations.

Findings

Characterization of algebras with smooth models via ramification divisors

Proof that the Brauer-Severi fibration is a flat morphism

Determination of irreducible components of fibers

Abstract

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.