Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories.Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic, growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has led to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.

Contents o. Introduction 1. The graded quotient ring of a graded prime ring of dimension two 2. The associated divisors 3. The stable image of the curve 4. Domains of dimension two which are generated in degree one 5. The case of an automorphism of infinite order 6. Applications to the structure of graded domains of dimension two 231 235 244 254 257 264 270

Let Hc be the rational Cherednik algebra of type A n−1 with spherical subalgebra Uc = eHce.Then Uc is filtered by order of differential operators, with associated graded ring grW is the n-th symmetric group. We construct a filtered Z-algebra B such that, under mild conditions on c:• the category B-qgr of graded noetherian B-modules modulo torsion is equivalent to Uc-mod;• the associated graded Z-algebra gr B has gr B-qgr ≃ Coh Hilb(n), the category of coherent sheaves on the Hilbert scheme of points in the plane.1991 Mathematics Subject Classification. 14C05, 32S45,16S80, 16D90.

Let X denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero. Denote by Dx the sheaf of differential operators on X, and D(X)=Γ(X,Dx), the ring of global differential operators on X. The following is established: THEOREM. D(X) is a finitely generated k‐algebra, and a noetherian ring. Furthermore, D(X) has a unique minimal non‐zero ideal J, and D(X)/J is a finite‐dimensional k‐algebra. Let X ˜ denoted the normalisation of X, and π: X ˜ →X the projection map. The main technique is to compare D(X) with D( X ˜ ). THEOREM. The following are equivalent: (i) π is injective, (ii) D(X) is a simple ring, (iii) D(X) is Morita equivalent to D( X ˜ ), (iv) the categories DX‐Mod and DX˜‐Mod are equivalent, (v) gr D(X) is noetherian, (vi) the global homological dimension of D(X) is 1. For higher‐dimensional varieties the techniques produce examples of varieties X for which D(X) is right but not left noetherian.

Let B(X, L, σ) be the twisted homogeneous coordinate ring of an irreducible variety X over an algebraically closed field k with dim X ≥ 2. Assume that c ∈ X and σ ∈ Aut(X) are in sufficiently general position. We show that if one follows the commutative prescription for blowing up X at c, but in this noncommutative setting, one obtains a noncommutative ring R = R(X, c, L, σ) with surprising properties. In particular:(1) R is always noetherian but never strongly noetherian.(2) If R is generated in degree one then the images of the R-point modules in qgr-R are naturally in (1-1) correspondence with the closed points of X. However, both in qgr-R and in gr-R, the R-point modules are not parametrized by a projective scheme.(3) qgr-R has finite cohomological dimension yet dim k H 1 (O R ) = ∞. This gives a more geometric approach to results of the second author who proved similar results for X = P n by algebraic methods.