International Mathematics Research Notices vol:1 pages:11-62
In this article, we consider surfaces that are general with respect to a three-dimensional toric idealistic cluster. In particular, this means that blowing up a toric constellation provides an embedded resolution of singularities for these surfaces. First we give a formula for the topological zeta function directly in terms of the cluster. Then we study the eigenvalues of monodromy. In particular, we derive a useful criterion to be an eigenvalue. In a third part, we prove the monodromy and the holomorphy conjecture for these surfaces.