Ring-Theoretic Blowing Down: I

Research output: Contribution to journalArticle

  • Authors:
  • D Rogalski
  • S. J. Sierra
  • John Stafford


One of the major open problems in noncommutative algebraic geometry is the classication of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). Earlier work of the authors classied 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.

Bibliographical metadata

Original languageEnglish
Pages (from-to)1465-1520
Number of pages56
JournalJournal of Noncommutative Geometry
Issue number4
Early online date15 Dec 2017
StatePublished - 2017