# Input-Output Descriptions (Singular version)

In the simplest case the input of the algorithm is an ideal, say J, over some base ring of characteristic zero generated by the variables, say x1, ..., xn. The ideal defines an affine scheme Z in the n-dimensional ambient affine space W corresponding to the base ring. It is also possible to start the algorithm using a nonsingular affine variety, say W, as the ambient variety (given by a polynomial ring and the ideal of the variety DEP) and the defining ideal J of the singular subvariety Z. In general, the user can start the algorithm with a weighted basic object (WBO) of his choice over a given affine chart. The WBO beside the ambient and singular varieties, can also contain exceptional divisors (E) with given multiplicities (a), with which they contribute to the singular variety. If the ambient variety is not an affine space, it is also necessary to define a regular parameter system and the partial derivations it gives rise to. For the details see the manual of the package.

The program computes an embedded desingularization f: Y -> W of Z, and produces a list of n-dimensional affine charts, that provide open covering for the variety Y. The full resolution, i.e. the complete list of charts (including the ones that are produced in the intermediate stages of the resolution), is available via the global list of rings DCR. The ith chart can be accessed by first setting the ith ring to be the basering (for instance def R = DCR[i]; setring(R);, then the variable DCH (which is a list) contains the data of the chart. For the detailed data structure please look up the manual of the package.

While the above output is provided for other algorithms to perform further computations, the HTML export of the resolution is intended for humans. Here is the summary of the data of chart-records (collected data of charts and basic objects) that are provided in the HTML output:

A chart U contains the following data:

VAR
a list of affine variables (generators of k[VAR]), in the localized version of the algorithm the local variables are distinguished with different grey background.
DEP
a list of polynomials from k[VAR] (the algebraic relations of VAR), k[U]= k[VAR]/< DEP >.
RSOP
a list of polynomials (representatives of elements from k[U], give rise to a system of regular parameters in every point of U),
PDER
a matrix of polynomials from k[VAR](being representatives of elements from k[U]), storing the partial derivatives of the generators of k[VAR] w.r.t. the RSOP members,
FOCUS
a list of polynomials from k[VAR], defining the subset of the chart which is not covered by charts with smaller index,
MAP
a list of polynomials from k[VAR], the images of the initial generators: x1, ..., xn in k[U].
IMAP
a list of rational functions from k(VAR'), where VAR' is the VAR member of the initial chart; these are the images of the generators of k[U] along the inverse of MAP,
TAGS
a list of lists of zeroes and ones tagging the IND entry. if an algebraic set is resolved, in the leaf-charts its strict transform can be defined by IND elements. Then, at the end of the resolution of the algebraic set a list that marks with 1-tags the defining IND elements is appended to TAGS. This has real usage in desingext and desingsep. This entry is nonempty only if there is already a resolution completed.
The stack contains weighted basic objects (WBO), tagged basic objects (TBO) and simple basic objects (SBO). Each TBO is created to have its singular locus equal to the equi-order locus of the previous WBO. Then each SBO is created to have the singular locus of the previous TBO restricted by the intersection of the maximal number of hypersurfaces from the EM entry of the TBO, whose intersection with the singular locus is not empty (ties are broken by choosing lex-maximal hypersurface tuples). Each WBO (except the one with largest dimension) is created to be one dimension less than the previous SBO (by coefficient object computation).

A weighted basic object contains the following data:

W
a list of pairwise different RSOP elements (internally these are indices to RSOP) defining the ambient variety W of the basic object,
J
the generators for the ideal of the basic object, polynomials from k[VAR], with rational exponents,
c
a rational number; the weighted order,
E
the list of equations (RSOP members) of exceptional divisors (internally these are also just indices to RSOP),
a
the weights of the exceptional divisors,
H
a list of equations (from RSOP) of exceptional divisors; that restrict the singular locus of the object (since otherwise they need not necessarily be normal crossing with the singular locus).
A tagged basic object contains the same data as a WBO, except the entries c and a, plus the entry:
EM
a list of equations (from RSOP) of exceptional divisors (the E- history).
A simple basic object contains the same data as a WBO, except the entries c and a.

The structure of the HTML-tree:

The charts form the tree of the resolution. The HTML output contains the vertices (i.e. charts) at their states before one of the three main operations (blowup, exchange, cover). In the tree the children of a vertex appear right below it. Please note that chronological information for a chart should not be inferred from the depth at which it appears in the tree, but by comparing its stratifying function value (displayed by default for each nodes) to other charts. Also please remember that the stratifying function is completely computed only at blowing ups (red vertices), while for other kind of vertices (green and blue) the printed stratifying function value is only partial. Moreover, if the "normal crossing test" is used with the original (i.e. non-localized) algorithm, the stratifying function values need not bee the correct ones. However, the computed morphism is correct also in this case.

For more information on the input-output, data structures, features and configuration of the package, please consult the manual.

ID: 19 Parent: 18 Tree
Covered by:
33 34
x(6) 1
PDER
VAR\RSOP x(6) x(2) x(5)
x(2) 0 1 0
x(5) 0 0 1
x(6) 1 0 0
DEP
MAP
x(1) x(2)*x(5)*x(6) x(2) x(2)*x(5)
IMAP
x(2) (x(2))/(1) (x(3))/(x(2)) (x(1))/(x(3))
FOCUS
x(5)
x(2)
x(2)*x(5)
x(2)*x(5)*x(6)
x(2)^2*x(5)^2*x(6)^2-x(2)^3*x(5)^2
STACK
WBO (dim: 3)
W
J
x(6)^2-x(2)1
c1
E,a
x(2)1
x(5)1
TBO (dim: 3)
W
J
x(6)^2-x(2)1
E
x(2)
x(5)
EM
x(2)
x(5)
SBO (dim: 3)
W
J
x(2)1
x(6)^2-x(2)1
E
H
x(2)
WBO (dim: 2)
W
x(2)
J
x(6)^22
c2
E,a
TBO (dim: 2)
W
x(2)
J
x(6)^22
E
EM
SBO (dim: 2)
W
x(2)
J
x(6)^22
E
Covered by:
33 34
x(6) 1
ID: 19 Parent: 18 Tree