Plumbing Graphs for Normal Surface-Curve Pairs

Abstract

Consider the set of surface-curve pairs $(X,\mathcal{C})$, where $X$ is a normal surface and $\mathcal{C}$ is an algebraic curve. In this paper, we define a family $\mathcal{F}$ of normal surface-curve pairs, which is closed under coverings, and which contains all smooth surface-curve pairs $(X, \mathcal{C})$, where $X$ is smooth and $\mathcal{C}$ has smooth irreducible components with normal crossings. We give a modification of W. Neumann’s definition of plumbing graphs, their associated 3-dimensional graph manifolds, and intersection matrices, and use this construction to describe rational intersection matrices and boundary manifolds for regular branched coverings.