The monodromy of a degenerating family of curves

Abstract

In this paper we analyze the homological monodromy of a degenerating family g: V-*D of complex curves over the disk, in particular developing a necessary and sufficient criterion for it to be of finite order. This criterion is in terms of what we call the number of cycles in the special fiber g-l(0). A one-dimensional analytic space contains cycles when its components intersect to form circular chains. We look at this in two cases, the special fiber of the degenerating family, and the exceptional locus in the resolution of a surface singularity. If the degenerating family is normal, that is, the special fiber is a reduced curve with at most ordinary double points, we show by a topological argument that the monodromy is the identity or of infinite order, the latter occuring precisely when g-l(0) contains cycles (Theorem 1). Mumford's semistable reduction theorem asserts that any degenerating family is dominated by a normal degenerating family (Proposition 2). The main result of this paper (Theorem 2) asserts that if the special fiber of an arbitrary degenerating family contains cycles, then its monodromy is of infinite order, and that its monodromy is of finite order precisely when the normal degenerating family dominating it contains no cycles. To prove this, we completely analyze dominating families (Proposition 1). Finally, we specialize to the case where the degenerating family consists of local curves, that is, the local monodromy of an analytic function f (x , y) of two variables. We show that it is of infinite order if the singularity f (x , y ) z" = 0 contains cycles for some n > 0, and that it is of finite order precisely when f ( x , y ) z N = 0 contains no cycles, where N is computed from the multiplicity sequence of f. I_~ [6] proved in the local case that the monodromy is of finite order when f is irreducible; A'Campo [I] provided another proof, and the first example of infinite order with f reducible. We present another example (Example 3). The methods of this paper allow one to easily decide whether the monodromy of any degenerating family of curves (local or not) is of finite order: One only need blow up at points of curves on the non-singular surface V, keeping track of multiplicities. Another criterion for infinite order by way of knot theory is presented in [3], and generalized in [13]. Throughout this paper we will use the term curve (surface) for a pure one (two) dimensional complex analytic space. We include the possibility that a curve or surface Z have boundary, namely that Z is a topological space with two subsets S z and B z such that (i) S z and B z are disjoint, (ii) Z S z is a smooth manifold with boundary B z, (iii) Z B z is a complex analytic space with singular