Monodromy of plane curves and quasi-ordinary surfaces

Abstract

In general the singular locus of such a surface is one-dimensional, with at most two components. A transverse slice x = C (where C is a small nonzero constant) cuts out a singular plane curve. The Milnor fiber of this curve undergoes a monodromy transformation when C loops around the origin; the action on its homology groups is called the vertical monodromy. In this article we show how to explicitly calculate this monodromy. Our formula is expressed recursively, by associating to our surface two related quasi-ordinary surfaces which we call its truncation S1 and its derived surface S′, and then expressing the vertical monodromy of S via the monodromies of S1 and of S′. As is well known, there is another fibration over a circle, called the Milnor fibration; here the action on homology is called the horizontal monodromy. In the course of working out our recursion for vertical monodromy, we have discovered what appears to be a new viewpoint about the horizontal monodromy, expressed in a similar recursion which again invokes the same two associated surfaces. In fact this recursion makes sense even outside the quasi-ordinary context, and thus we have found a novel way to express the monodromy associated to the Milnor fibration of a singular plane curve. We begin by working out this situation, to motivate our later setup and to provide a model for the more elaborate calculation. As a corollary to our formulas, we have found that from the vertical monodromies (one for each component of the singular locus), together with the surface monodromy formula worked out in [11] and [4], one can recover the complete set of characteristic pairs of a quasi-ordinary surface. Since these data depend only on the embedded topology of the surface, we thus have a new proof of Gau’s theorem [3] in the 2-dimensional case. As another application, we can employ a theorem of Steenbrink [13] (extended to the non-isolated case by M. Saito [12]) which relates the horizontal and vertical monodromies to the spectrum of the surface and to the