Non-commutative Mori contractions and $\PP^1$-bundles

TL;DR

This paper develops a method to construct maps from non-commutative schemes to curves, enabling the study of non-commutative Mori contractions and $ ext{P}^1$-bundles, with applications to non-commutative surface classification.

Contribution

It introduces a new approach using Artin-Zhang's Hilbert schemes to construct non-commutative Mori contractions and analyzes non-commutative $ ext{P}^1$-bundles and ruled surfaces.

Findings

Non-commutative $ ext{P}^1$-bundles are smooth and have well-behaved Hilbert schemes.

Non-commutative ruled surfaces exemplify non-commutative Mori contractions.

The Serre functor for non-commutative $ ext{P}^1$-bundles is explicitly computed.

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 $\PP^1$-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.