We give an algorithm that, for a given value of the geometric genus p g , computes all regular product-quotient surfaces with abelian group that have at most canonical singularities and have canonical system with at most isolated base points. We use it to show that there are exactly two families of such surfaces with canonical map of degree 32. We also construct a surface with q = 1 and canonical map of degree 24. These are regular surfaces with p g = 3 and base point free canonical system. We discuss the case of regular surfaces with p g = 4 and base point free canonical system.

In this paper some numerical restrictions for surfaces with an involution are obtained. These formulas are used to study surfaces of general type S with p g = q = 1 having an involution i such that S/i is a non-ruled surface and such that the bicanonical map of S is not composed with i. A complete list of possibilities is given and several new examples are constructed, as bidouble covers of surfaces. In particular the first example of a minimal surface of general type with p g = q = 1 and K 2 = 7 having birational bicanonical map is obtained.

Abstract. This paper describes how to compute equations of plane models of minimal Du Val double planes of general type with p g = q = 1 and K 2 = 2, . . . , 8. A double plane with K 2 = 8 having bicanonical map not composed with the associated involution is also constructed. The computations are done using the algebra system Magma.

We give a list of possibilities for surfaces of general type with pg = 0 having an involution i such that the bicanonical map of S is not composed with i and S/i is not rational. Some examples with K 2 = 4, . . . , 7 are constructed as double coverings of an Enriques surface. These surfaces have a description as bidouble coverings of the plane.

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