  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.
