Ruled surfaces and generic coverings

Abstract

In this paper we study normal singularities of complex analytic surfaces. In particular, since a normal surface has only isolated singularities (see, e.g., [10]), we restrict our attention to germs (S, s) where S is a connected normal surface, s ∈ S and S s is non-singular. We begin by setting some standard notation. A normal generic covering (S,π) (ngc in the sequel) is a finite holomorphic map π :S → C2 from a connected normal surface S to the complex plane C2, which is an analytic covering branched over a curve B ⊂C2, such that the fiber over a smooth point of B is supported on degπ − 1 distinct points. A ngc is called smooth if S is non-singular. Two ngcs (S1,π1), (S2,π2) are called (analytically) equivalent if there exists an isomorphism φ :S1 → S2 such that π1 = π2 ◦ φ. In the sequel, we will consider equivalent ngcs to be the same covering. The main interest in ngcs comes from the well-known fact that, by Weierstrass preparation theorem, given an analytic surface S ⊂ C, a generic projection S π −→ C2 is (at least locally, in order to insure degπ < ∞) a ngc branched over a curve (see [4]). Classically, one would like to reconstruct every ngc starting from downstairs data (i.e. in C2, like the branch curve B). Over a non-singular point of B , π is locally (in S) equivalent to the map of the complex plane to itself which takes (x, y) to (x, y) with a = 1,2. The main point is then to study germs of ngcs where the branch curve is a singular germ of a plane curve. We can then restrict to the case in which the branch locus B has only one singular point, which we may assume to be the origin O . For a fixed curve B there are three natural problems related to ngcs: the existence problem (there exists a ngc branched over B?), the smoothness problem (there exists a smooth ngc branched over B?), and the uniqueness problem (under which hypothesis is the covering unique? This is related to a conjecture of Chisini, see [8]).