  We give a method for constructing maps from a non-commutative scheme to a
commutative projective curve. With the aid of Artin-Zhang's abstract Hilbert
schemes, this is used to construct analogues of the extremal contraction of a
$K$-negative curve with self-intersection zero on a smooth projective surface.
This result will hopefully be useful in studying Artin's conjecture on the
birational classification of non-commutative surfaces. As a non-trivial example
of the theory developed, we look at non-commutative ruled surfaces and, more
generally, at non-commutative $\PP^1$-bundles. We show in particular, that
non-commutative $\PP^1$-bundles are smooth, have well-behaved Hilbert schemes
and we compute its Serre functor. We then show that non-commutative ruled
surfaces give examples of the aforementioned non-commutative Mori contractions.
