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Configuring special techniques

There are a few available special representation or computation techniques which are not used by default, however, in certain examples they might lead to the faster computation of the resolution.


By the configuration variable usefocus one can control if in each chart of the covering a list of equations, called FOCUS, is maintained to eliminate redundant charts. By default it is true, and one can deactivate the feature by setting

  desingcfg[usefocus] := false;

Algebraic dependencies

The configuration variable useDEP2 (default: false) controls the method which allocates new variables for inverses of functions from the coordinate ring of the chart.

If the variable is false, each invertible function gets its own variable, which denotes its inverse in the coordinate ring of the chart. This leads to the increase of variables, which is unfortunate from the point of view of Gröbner basis computations. On the other hand the polynomials with which we have to work are simpler than in the other case.

If the variable is true, there is a single variable allocated for the product of the invertible functions from the coordinate ring of the chart. In this case the number of variables can be kept constant as new invertible functions are introduced in the cover transformations. On the other hand the polynomials with which we have to work with can grow much larger and denser than in the other case.

Covering blowing up varieties

The configuration variable skewcover (default: false) controls the strategy by which the blowing up varieties are covered in blowup transformations.

If the variable is false, since the blowing up centers, locally over the charts, are always defined by regular parameters, the blowing up variety can be covered by affine charts in a straightforward manner. However, the dimension of the ambient affine space (i.e. the number of generators of the coordinate ring) of the new charts may in general increase.

If the variable is true, a sophisticated covering strategy is applied, in which the defining equations of the blowing up center are replaced by others, which allow us to define affine patches of the blowing up variety which can be embedded in affine spaces of the same dimension as the ambient affine space of the original chart. It can be proved that only finitely many charts are needed to completely cover the blowing up variety.

The configuration variable rotationrange (default: 10) also effects this feature, setting the upper bound for the coefficient of the polynomials used in the random rotations of the coordinate systems over the charts.

If skewcover is true, the program also sets useDEP2 to true for the time of the computations.

This feature is of experimental nature and need not provide superior performance to the standard covering techniques. The `desing/adjoints`, `desing/dualgraph` and `desing/blunify` procedures do not work when the resoltuion is computed using this technique due to data incompatibility reasons. This covering strategy supresses the normal crossing test (section *), a cooperative application of the two techniques might be implemented later.

The results related to this covering strategy are planned to be described in an upcoming paper.

Testing normal crossing

The configuration variable usenctest (default: false) controls the way a component of the Villamayor stratifying function is computed.

If the variable is false, the Villamayor function is computed as it is defined in the paper [EV00].

If the variable is true, the step when the maximum locus of the function is restricted to the intersection of the maximal number of exceptional divisors from E- (see Definition 4.14 on page 160 of [EV00]) with certain prescribed properties, is computed differently. This time, instead of letting all the exceptional divisors which have the required properties to restrict the maximum locus, we select only those ones which violate the normal crossing property. This way, the dimension of the blowing up centers can be increased, and the resolution can often be simplified greatly.

The results are planned to be described in an upcoming paper.

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