Noncommutative Tsen's theorem in dimension one

TL;DR

This paper establishes necessary and sufficient conditions for noncommutative genus zero curves over a field to be noncommutative P^1-bundles, extending Tsen's theorem to a noncommutative, one-dimensional setting and characterizing arithmetic noncommutative curves.

Contribution

It introduces a noncommutative version of Tsen's theorem for genus zero curves and characterizes arithmetic noncommutative projective lines and curves.

Findings

Characterization of noncommutative P^1-bundles over genus zero curves

Equivalence between arithmetic noncommutative projective lines and noncommutative curves

Resolution of specific problems posed by D. Kussin

Abstract

Let k be a field. In this paper, we find necessary and sufficient conditions for a noncommutative curve of genus zero over k to be a noncommutative P^1-bundle. This result can be considered a noncommutative, one-dimensional version of Tsen's theorem. By specializing this theorem, we show that every arithmetic noncommutative projective line is a noncommutative curve, and conversely we characterize exactly those noncommutative curves of genus zero which are arithmetic. We then use this characterization, together with results regarding arithmetic noncommutative projective lines, to address some problems posed by D. Kussin.