Non-commutative Mori contractions and [formula omitted]-bundles

Abstract

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 P1-bundles. We show in particular, that non-commutative P1-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.