Both resolution graphs and splice diagrams are labelled graph-like diagrams used to encode geometric data closely related to some resolution of singularities procedure in algebraic geometry. They are commonly used to visualise this data. Of course, there are other tools in Magma, Puiseux expansions for instance, which can be used if preferred. A typical example is when a configuration of curves on a surface is the given data. In this case, dual graph of the configuration has vertices corresponding to the individual curves and edges corresponding to their intersections. The vertices may be labelled with the selfintersections of the corresponding curves and possibly also with the multiplicities with which the curves appear in the configuration.
This chapter discusses two different enhanced graph types: GrphRes for resolution graphs and GrphSpl for splice diagrams. Neither of them is literally a graph in Magma; in particular, functions taking graphs as argument cannot be applied directly to objects defined here. Instead they work by having a directed graph, referred to as the underlying graph, as a primary attribute and by caching other data (which is typically associated to particular vertices and edges of the graph) in sequences as secondary attributes. There are also vertex types which allow the convenient idiom of Magma's graph package to be used. Note, however, that unlike other graph types, these do not have edge types.
Graph surgery routines cannot be used directly since they must manage both the underlying graph and the associated data. A collection of appropriate surgery functions, those used in resolution routines, have been provided in this context; they are usually brief, simply concatenating attribute data whenever it is present on both sides.
Functions to recover data related to the graph, whether globally or at a particular vertex or edge, are also provided. Thus the user does not access labels of the graph but rather uses the intrinsics listed later in this chapter. Of course, usually these do no more than unload an attribute.