ar X iv : 0 80 5 . 45 13 v 1 [ m at h . A G ] 2 9 M ay 2 00 8 Involutions on surfaces with p g = q = 1

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 pg = 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 pg = q = 1 and K 2 = 7 having birational bicanonical map is obtained. 2000 Mathematics Classification: 14J29.