In this paper we generalize some classical birational transformations to the non-commutative case. In particular we show that 3-dimensional quadratic Sklyanin algebras (non-commutative projective planes) and 3-dimensional cubic Sklyanin algebras (non-commutative quadrics) have the same function field. In the same vein we construct an analogue of the Cremona transform for non-commutative projective planes.

Abstract. In this note we consider a notion of relative Frobenius pairs of commutative rings S/R. To such a pair, we associate an N-graded R-algebra Π R (S) which has a simple description and coincides with the preprojective algebra of a quiver with a single central node and several outgoing edges in the split case. If the rank of S over R is 4 and R is noetherian, we prove that Π R (S) is itself noetherian and finite over its center and that each Π R (S) d is finitely generated projective. We also prove that Π R (S) is of finite global dimension if R and S are regular.

Recently de Thanhoffer de Völcsey and Van den Bergh classified the Euler forms on a free abelian group of rank 4 having the properties of the Euler form of a smooth projective surface. There are two types of solutions: one corresponding to double-struckP1×double-struckP1 (and non‐commutative quadrics), and an infinite family indexed by the natural numbers. For m=0,1 there are commutative and non‐commutative surfaces having this Euler form, whilst for m⩾2 there are no commutative surfaces. In this paper, we construct sheaves of maximal orders on surfaces having these Euler forms, giving a geometric construction for their numerical blowups.

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