Explanation and remarks for the output of the dualgraph routine

What is "one" exceptional divisor?: Since the algorithm does not decompose blowing up centers into irreducible components we also do not do the costly operation in the dualgraph computation. Therefore we use the reasonable convention that 'one (collection of) exceptional divisor(s)' consists of all the exceptional components which were created by a single blowing up prescribed by Villamayor's stratifying function. That is, the stratifying function values determine which components are to be taken in a collection of exceptional divisors. In the case when the center arises in a monomial object, the last component of the stratifying function (the list of the indices of the chosen exceptional divisors whose intersection defines the blowing up center) contains local indices, thus, in general, they cannot be used to detect patching of centers in different charts. In such cases the global indices of exceptional divisors are reconstructed and the true stratifying function value is computed (now using global indices for exceptional divisors instead of local ones).

Notation of the exceptional divisors: An element of the list of exceptional divisors consists of the symbolic name (e.g. E₁) of the divisor and the list of charts in which it appears (i.e. in which its restriction is nonempty), together with the index of the E element of the basic object with largest dimension which defines it.

Description of blowing up centers: The third column contains the value of the stratifying function along the center of the blowing up which created the exceptional divisor (defined on the same row). Under the function value appears the list of charts in which the center appears (i.e. in which its restriction is nonempty). Under that one finds the list of indices of ("global") exceptional divisors which contain the blowing up center (at the stage of the resolution when it was computed). The last datum is the dimension of the center.

Remark: The boldface chart IDs mark the charts of the final ambient variety (in which the resolution problem is solved (modulo hypersurface blowing ups)).

What does the stratifying function value consist of?: We used the definition of the function which can be found in the paper: A Course on Constructive Desingularization and Equivariance, S. Encinas and O. Villamayor, in Resolution of Singularities, A research textbook in tribute to Oscar Zariski editors H. Hauser, J. Lipman, F. Oort, A. Quirós, Progress in Mathematics 181, Birkhäuser Boston, 2000.

The values of the subfunctions which are computed in different dimensions are separated by parentheses or brackets. The values of the generic (nontrivial) branch of the function appear in brackets, while the values of the monomial and good-point branches in parentheses (please recall that any value in parentheses is smaller than any other in brackets (provided that they measure resolution problems of the same dimension)).

The order components of the values should be understood as rational orders computed for powered ideals (see, for instance, Two computational techniques for singularity resolution, G. Bodnár and J. Schicho, Journal of Symbolic Computation, 32(1-2): 39-54, 2001, or Algorithmic Resolution of Singularities, G. Bodnár, PhD thesis, Johannes Kepler University, RISC-Linz, 2000), which are just the c components of weighted basic objects. In the same context, the exponents in the a functions can also well be rational numbers.

The last component of the parenthesized function values stands for the list of exceptional divisors defining the blowing up center in the monomial case. The numbers in these lists index the global exceptional divisor list (therefore they can be different from the numbers that appear in the chart data or in the chart tree).

Asterisks mark the case when the codimension-one part of the maximum stratum is not empty, thus the maximal function value is 'infinity'.

Notation of the hyperbonds: A hyperbond is defined by the indices of the vertices which it connects. A pair defines an edge, a triple a face, a quadruple a solid, etc.

The list does not contain those hyperbonds of lower dimension which are implied by higher dimensional ones. That is, if there is a face between three vertices, say [1,2,3], then the edges between all the pairs of the vertices, [1,2],[1,3],[2,3], are implied by the face .