Ring-theoretic blowing down II: Birational transformations

TL;DR

This paper develops methods for birational transformations in noncommutative algebraic geometry, including explicit constructions and analogues of classical transformations, advancing the classification of noncommutative surfaces.

Contribution

It introduces explicit noncommutative birational transformations, including a noncommutative Cremona transform, building on previous work and connecting different noncommutative surface models.

Findings

Van den Bergh's quadrics derived from Sklyanin algebra via blowups and blowdowns

Constructed a noncommutative analogue of the classical Cremona transform

Extended earlier work on noncommutative surface transformations

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). In a companion paper the authors described a noncommutative version of blowing down and, for example, gave a noncommutative analogue of Castelnuovo's classic theorem that lines of self-intersection (-1) on a smooth surface can be contracted. In this paper we will use these techniques to construct explicit birational transformations between various noncommutative surfaces containing an elliptic curve. Notably we show that Van den Bergh's quadrics can be obtained from the Sklyanin algebra by suitably blowing up and down, and we also provide a noncommutative analogue of the classical Cremona transform. This extends and amplifies earlier work of Presotto and Van den Bergh.