Zeta function and value semigroup of complex plane curve singularities

Abstract

In the work by A’Campo and Oka (1996), Tschirnhausen resolution towers arose as a nice tool when studying complex irreducible plane curve singularities. Using this tool combinatorially, we can revisit the extended simplified resolution graph or the toric resolution tree developed previously by the second author. We restate that the graph can help describe a resolution, namely, the monodromy and Alexander zeta functions of a complex reducible degenerate plane curve singularity. Furthermore, we show the relation between the monodromy zeta function and the value semigroup in the irreducible case, and we also give some comments on the reducible case.