  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). Earlier work of the authors classified 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 blown-down 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.
