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 x_{1}, ..., x_{n}. The
ideal defines an affine scheme Z in the
ndimensional 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 ndimensional
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 chartrecords (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: x_{1}, ...,
x_{n} 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 leafcharts 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 1tags
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 equiorder 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 lexmaximal 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 HTMLtree:
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. nonlocalized) 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 inputoutput, data structures,
features and configuration of the package, please consult the
manual.


ID: 19 
Parent: 18

Tree 
 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(3)  x(2)*x(5) 

IMAP 
x(2)  (x(2))/(1) 
x(5)  (x(3))/(x(2)) 
x(6)  (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)^2x(2)^3*x(5)^2 

STACK 
WBO (dim: 3) 
W  
J 
x(6)^2x(2)  1 
c  1 
E,a 
x(2)  1 
x(5)  1 
TBO (dim: 3) 
W  
J 
x(6)^2x(2)  1 
E 
x(2) 
x(5) 
EM 
x(2) 
x(5) 
SBO (dim: 3) 
W  
J 
x(2)  1 
x(6)^2x(2)  1 
E  
H 
x(2) 
WBO (dim: 2) 
W 
x(2) 
J 
x(6)^2  2 
c  2 
E,a  
TBO (dim: 2) 
W 
x(2) 
J 
x(6)^2  2 
E  
EM  
SBO (dim: 2) 
W 
x(2) 
J 
x(6)^2  2 
E  
 ID: 19 
Parent: 18

Tree 

