Abstract. One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). Earlier work of the authors classified the connected graded noetherian subalgebras of Sklyanin algebras using a noncommutative analogue of blowing up. In order to understand other algebras birational to a Sklyanin algebra, one also needs a notion of blowing down. This is achieved in this paper, where we give a noncommutative analogue of Castelnuovo's classic theorem that (−1)-lines on a smooth surface can be contracted. The resulting noncommutative blowndown algebra has pleasant properties; in particular it is always noetherian and is smooth if the original noncommutative surface is smooth.In a companion paper we will use this technique to construct explicit birational transformations between various noncommutative surfaces which contain an elliptic curve.

Given a projective scheme X over a field k, an automorphism σ : X → X, and a σ-ample invertible sheaf L, one may form the twisted homogeneous coordinate ring B = B(X, L, σ), one of the most fundamental constructions in noncommutative projective algebraic geometry. We study the primitive spectrum of B, as well as that of other closely related algebras such as skew and skew-Laurent extensions of commutative algebras. Over an algebraically closed, uncountable field k of characteristic zero, we prove that the primitive ideals of B are characterized by the usual Dixmier-Moeglin conditions whenever dim X ≤ 2.

Given Z-graded rings A and B, we ask when the graded module categories gr-A and gr-B are equivalent. Using Z-algebras, we relate the Morita-type results of Áhn-Márki and del Río to the twisting systems introduced by Zhang, and prove, for example: Theorem If A and B are Z-graded rings, then: (1) A is isomorphic to a Zhang twist of B if and only if the Z-algebras A = i, j∈Z A j−i and B = i, j∈Z B j−i are isomorphic. (2) If A and B are connected graded with A 1 = 0, then gr-A gr-B if and only if A and B are isomorphic. This simplifies and extends Zhang's results.

Abstract. One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative surfaces, and this paper resolves a significant case of this problem.Specifically, let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field k and assume that S is not a finite module over its centre. (This algebra corresponds to a generic noncommutative P 2 .) Let A = i≥0 A i be any connected graded k-algebra that is contained in and has the same quotient ring as a Veronese ring S (3n) . Then we give a reasonably complete description of the structure of A. This is most satisfactory when A is a maximal order, in which case we prove, subject to a minor technical condition, that A is a noncommutative blowup of S (3n) at a (possibly non-effective) divisor on the associated elliptic curve E. It follows that A has surprisingly pleasant properties; for example it is automatically noetherian, indeed strongly noetherian, and has a dualizing complex.

We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element g of degree 1, T/gT is a twisted homogeneous coordinate ring of an elliptic curve E at an infinite order automorphism). Given an effective divisor d on E whose degree is not too big, we construct a blowup T(d) of T at d and show that it is also an elliptic algebra. Consequently it has many good properties: for example, it is strongly noetherian, Auslander-Gorenstein, and has a balanced dualizing complex. We also show that the ideal structure of T(d) is quite rigid. Our results generalise those of the first author. In the companion paper "Classifying Orders in the Sklyanin Algebra", we apply our results to classify orders in (a Veronese subalgebra of) a generic cubic or quadratic Sklyanin algebra.Comment: 39 pages. Minor changes from previous version. The final publication is available from Springer via http://dx.doi.org/10.1007/s10468-014-9506-