The monodromy of reducible plane curves

Abstract

Let f : C "+1~ C be a polynomial function such that f ( 0 ) = 0 and f has an isolated singularity at 0. The local geometry of f 1 (0) near 0 is completely described by the algebraic link L 2"-1 of the singularity, where L 2" l = f l ( 0 ) n S 2"+1 is a smooth (n-2)-connected submanifold of the small E-sphere S 2"+i about the origin [13]. Moreover, the link complement X = S 2 " + I L fibers over S 1 with fiber F2"~_S"v ... v S". Let X denote the infinite cyclic covering space of X associated with the epimorphism H 1 ( X ) ~ HI(S 1) induced by the fibration. The action of the multiplicative generator t of 171(S 1) on H,(F;Z) is called the local algebraic monodromy transformation of f at O. Since )( ~ F x R, t can be viewed as the generator of the infinite cyclic covering translation group, and the algebraic monodromy is an automorphism t: H,( ) ) ; Z ) ~ Hn0); Z). The polynomial invariants of H, (X; Z) as a A = Z [t, t 1] module are therefore invariants of the monodromy. For example, the Alexander polynomial A t (t) is the characteristic polynomial of the monodromy. We are interested in the following problem: Given ) , determine whether the algebraic monodromy is of finite or infinite order. The problem is most amenable to solution in the case n= 1, because the geometry is completely understood [3,9, 15]. In the classical case (n=l) , f l ( 0 ) is called a plane curve, and the algebraic link L 1 has r components, one corresponding to each of the branches of f 1(0) at the origin. Each branch corresponds to a distinct analytically irreducible factor of f which maps 0 to 0. Each component of L is an iterated torus knot, and both the iteration on each component and the linking among the various components is completely specified by the Puiseux developments [1, i0] corresponding to the branches. In the case that f is analytically irreducible (r = 1), f always has finite monodromy [10, 1, 19]. If r>2 , the monodromy can be infinite [1, 20]. This paper