General Minimal Surfaces Three Times A Model For Mapping

This paper studies minimal surfaces of general type with pg > 3 and canonical maps 3-to -1 onto a surface of minimal degree pg - 2 in Ppg-1. We have constructed/reconstructed many kinds of the surfaces with these properties, by using Tan's theory about triple cover and the method on the resolution of the singularities from the triple covering data directly. And these examples may help us to know more about the structure of these kinds of surfaces.In order to construct the above examples, we establish a necessary and sufficient condition on the triple covering data. Usually this condition will be known only after the resolution of the singularities is done. In this paper, however, we find a new condition which can be checked easily from the triple covering data.In addition, in section 2 we introduce a local coordinate system on Hirzebruch surface Fe, under which each divisor on Fe can be written in a nice form. We provide a new proof of the existence of such a local coordinate system.