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

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

2009-04-10

arxiv.org/abs/0904.1717v1

Mathematics

Algebraic Geometry

49 pages

Scientific paper

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.

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