  In this paper we propose an approach to investigate the canonical rings of
surfaces of general type whose canonical system has isolated base points and
yields a birational map onto its image.
  We apply then the method in the concrete case of surfaces with $K^2=7$ and
$p_g=4$, where we give three different descriptions of the canonical ring
(under the presence of base points).
  The main result in the paper consists in showing an explicit deformation of
the above canonical rings to the canonical rings of surfaces whose canonical
system has no base points (the latter were already classified by the second
author). That such a deformation should exist was claimed by F. Enriques and
holds true by abstract deformation theory and dimension counts. Our deformation
is explicitly gotten by writing the equations of the canonical ring as the
(4x4)-Pfaffians of an antisymmetric and extrasymmetric (6x6)-matrix.
  Deforming the matrix, we get, for $t \neq 0$, that the Pfaffians reduce to
the (4x4)-Pfaffians of an antisymmetric (5x5)-matrix. This type of deformation
of a Gorenstein ring of codimension 4 to a Gorenstein ring of codimension 3 had
already been introduced by Duncan Dicks and Miles Reid.
