Canonical rings of surfaces whose canonical system has base points

Abstract

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.