  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.
