On normal surface singularities and a problem of Enriques

We construct families of normal surface singularities with the following pr operty: given any flat projective connected family V --> B of smooth, irred ucible, minimal algebraic surfaces, the general singularity in one of our f amilies cannot occur, analytically, on any algebraic surfaces which is bira tionally equivalent to a surface in V --> B. In particular this holds for V --> B consisting of a single rational surface, thus answering negatively t o a long standing problem posed by F. Enriques. In order to prove the above mentioned results, we develop a general, though elementary, method, based on the consideration of suitable correspondences, for comparing a given fam ily of minimal surfaces with a family of surface singularities. Specificall y the method in question gives us the possibility of comparing the paramete rs on which the two families depend, thus leading to the aforementioned res ults.