Non-commutative Mori contractions andP1-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 P 1 -bundles. We show in particular, that non-commutative P 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.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.This part is primarily concerned with addressing the question, "Which quasi-schemes are the noncommutative analogues of smooth proper varieties?". We start in section 2 with recounting the notion of base change and Hilbert schemes for categories developed in [AZ01]. Hilbert schemes will be a fundamental tool for us. In section 3, we propose a definition of a non-commutative smooth proper dfold based on various geometric conditions which we impose on a quasi-scheme Y . The list of hypotheses is rather long so in section 4, we show that in the projective case of Y = Proj A, these conditions do follow from hypotheses on A that other authors have studied in the past. Background on base change and Hilbert schemesLet R be a commutative k-algebra. Section B of [AZ01] is devoted to the notion of base change from k −→ R for arbitrary categories. In this section, we review their results and generalise their notion of R-objects to X-objects where X is a separated scheme. We will always work over a field k as opposed to more general commutative rings as in [AZ01]. This simplifies the treatment somewhat, since then any Grothendieck category has k-flat generators and the subtleties involved with defining tensor products disappear.Let Y be a quasi-scheme defined over the ground field k as always.and insisting M ⊗ R − is right exact and commutes with direct sums [AZ01, proposition B3.1]. We may thus define M to be R-flat (or just flat) if this functor is also left exact [AZ01, section C.1]. Also, given a morphism of commutative k-algebras R −→ S, there is a base change functor Mod Y R −→ Mod Y S . With this notion of base change, Grothendieck's theory of flat descent holds [AZ01, theorem C8.6].The categories Mod Y R thus form a stack and this allows us to do base change via arbitrary separated schemes. We copy the definition from that of quasi-coherent modules on a stack (see for example [Vistoli, Appendix]). Let X be a separated scheme on which we put the small Zariski site. An X-object M in Mod Y is the data of an R-object M R ∈ Mod Y R f...

Species and non-commutative P1's over non-algebraic bimodules

“…This follows from the usual proof for classical Serre duality [10] and the cohomology of line bundles computations in Section 7. The reader can check [4] for details.…”

The birational geometry of noncommutative surfaces

“…Again, the corresponding moduli stack is itself locally Noetherian, so the claims follow in general. Moreover, over a Noetherian base, the category is again locally Noetherian; in this case, the argument from the literature [15] can be adapted with no difficulty (since they were already considering constant families over general Noetherian k-algebras). For technical reasons, we will define a "split" family of quasi-ruled surfaces to be a surface obtained from the above data in which we have also marked points x i ∈ C i (R).…”

“…In [15], a definition was given for a "non-commutative smooth proper d-fold", and it was shown there that ruled surfaces (more precisely, quasi-ruled surfaces in which the two curves are isomorphic) satisfy their definition (with d = 2, of course). Our objective is to show that this continues to hold for iterated blowups of arbitrary quasi-ruled surfaces.…”

“…In [15], Chan and Nyman proposed a definition of "proper" noncommutative surfaces and established that noncommutative ruled surfaces satisfied their definition. It should not be too surprising, therefore, that we can show that their axioms are satisfied by any rationally quasi-ruled surface.…”

Non-commutative $\mathbb{P}^{1}$-bundles over commutative schemes

“…In [8] the authors attack the reverse question. They generalize a standard characterization of ruled surfaces [12] to the non-commutative case.…”