Non-commutative Mori contractions and P-bundles

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 P-bundles. We show in particular, that non-commutative P-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. Throughout, all objects and maps are assumed to be defined over some algebraically closed base field k. The first author was supported by an ARC Discovery Project grant.