Go backward to Implementation Aspects of the Resolution Algorithm: Blowing Up Along Nontrivial Centers Go up to Winter Semester '98 Go forward to Equivalence of Definitions of Adjoints (talk 1)

Implementation Aspects of the Resolution Algorithm:
Eliminating Chart Redundancies &
Efficient Handling of Hypersurfaces Restricting the Singular Loci of Basic Objects

Talker: Gábor Bodnár  

Experiments has shown that the chart propagation, caused by the cover and the blowing up operations, lead to redundant tasks for the algorithm. The simple and ingenious solution came from J. Schicho. This solution requires some additional bookkeeping whenever new charts are produced, and achieves global minimality. The main idea behind the solution is to keep track in each chart the area that is already covered by previously created charts, more precisely, the complementer area, that is "interesting" in this chart. Then in the cover and blowing up operations those newly created charts which have empty "interesting area" are redundant, and are dropped. Unfortunately this solution cannot restrict the attention to only the "interesting area" since it might not contain the singular locus, but gives us a shortcut in cases when the singular locus and this area is disjoint.

In this talk we present the necessary extension of chart describing data to make the above mentioned method work. We discuss the usage of the information we gain by the new data member, and describe the modifications of the algorithms cover, and blowup.

Part 2: The Villamayor stratification function depends not only on the current singular locus of a basic object, but also on the "history" of the exceptional divisors of previous blowing ups. This means for the resolution algorithm that the ideals of the basic objects have to be modified under certain conditions, for instance to restrict the singular locus to the intersection of exceptional divisors to deal first with the singular points lying within this area. This adds complications to the computations and extends the computation time. With additional bookkeeping we can avoid modifying the ideal in any situation, making the computation faster.

In this talk we present the circumstances under which the ideals of basic objects have to be modified, then we describe the necessary modifications in the data of basic objects and in the resolution algorithm.