ℚ-Orbital Surfaces

Abstract

We want to extend the numerical theory of Galois coverings of algebraic surfaces to arbitrary surface coverings. For this purpose it is necessary to extend the notion of orbital surfaces. In the Galois theory orbital surfaces, orbital curves and points can be expressed by means of divisors and singularities. These are quite classical objects. The classical language does not work nicely in the general theory of surface coverings. Here we have to introduce sums of orbital curves and orbital points with rational coefficients. They burst open the old framework of divisor language. For a first motivation we remember to the idea of Riemann surfaces. Riemann constructed these surfaces by means of several exemplars of the complex affine line 𝔸1 = 𝔸1 (ℂ) = ℂ patched together along cuts. Then he transported geometry and complex analysis from ℂ to Riemann surfaces. This fruitful work has been extended by many outstanding mathematicians and it is not finished until these days. For algebraic surfaces we converse in some sense the idea. Let X, Y be algebraic surfaces, say smooth, complex, compact, and f : X → Y a finite covering. The branch locus B f of f is a (reduced) divisor on Y. For a simple illustration we refer to see Figure 6.1.1