Some Noncommutative Minimal Surfaces

TL;DR

This paper classifies certain noncommutative surfaces, showing that the generic noncommutative projective plane and related analogues are minimal models, and characterizes their maximal orders and overrings.

Contribution

It establishes strong minimality conditions for noncommutative projective surfaces like the Sklyanin algebra and Van den Bergh quadrics, and describes their maximal orders and overrings.

Findings

The generic noncommutative projective plane is minimal among certain graded algebras.

Maximal orders containing these surfaces are isomorphic to the original algebra.

Overrings of elliptic algebras are obtained by blowing down finitely many lines.

Abstract

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is to determine the minimal models within any birational class. In this paper we show that the generic noncommutative projective plane (corresponding to the three dimensional Sklyanin algebra R) as well as noncommutative analogues of P^1 x P^1 and of the Hirzebruch surface F_2 (arising from Van den Bergh's quadrics R) satisfy very strong minimality conditions. Translated into an algebraic question, where one is interested in a maximality condition, we prove the following theorem. Let R be a Sklyanin algebra or a Van den Bergh quadric that is infinite dimensional over its centre and let A be any connected graded noetherian maximal order containing R, with the same graded quotient ring as R. Then, up to taking Veronese rings, A is isomorphic to R. Secondly, let T be an elliptic algebra (that is, the coordinate ring of a noncommutative surface containing an elliptic curve). Then, under an appropriate homological condition, we prove that every connected graded noetherian overring of T is obtained by blowing down finitely many lines (line modules).