Some Invariants for Conies and Their Applications

Abstract

1. Compared with the case of lines, very little is known about the question: what kind of configurations of conies are possible in the (complex) projective plane. In this note we attempt to introduce some analytic invariants to control the intersection behavior of conies; if one wants to study the configurations of conies with a prescribed contact behavior, these invariants might serve as the natural parameters which they depend on. In fact we will describe the parameter space (the moduli) for some elementary configurations which may often appear as parts of some more complicated ones. We will also clarify how the invariants behave under duality. As a by-product we obtain three families of quartic surfaces in P3 with three singular points of type A3 and seven ordinary double points (A}): two of them are dual to each other and the remaining one is self-dual. This work was originally motivated by the problem of finding interesting abelian covers of P2 branched over several conies. The author would like to express his sincere thanks to Professor Hirzebruch for the introduction to this problem. He is also very grateful to Professor Brieskorn for the discussion which led to the generalization of the invariants.