Automated Deduction in Geometry 4th International Workshop, ADG 2002
This book constitutes the thoroughly refereed post-proceedings of the 4th International Workshop on Automated Deduction in Geometry, ADG 2002, held at Hagenberg Castle, Austria in September 2002. The 13 revised full papers presented were carefully selecte
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Abstract. Given a finite set of varieties in some nonsingular affine variety W . They are normal crossing if and only if at every point of W there is a regular system of parameters such that each variety can be defined locally at the point by a subset of this parameter system. In this paper we present two algorithms to test this property. The first one is developed for hypersurfaces only, and it has a straightforward structure. The second copes with the general case by constructing finitely many regular parameter systems which are “witnesses” of the normal crossing of the varieties over open subsets of W . The ideas of the methods are applied in a computer program for resolution of singularities.
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Introduction
The notion of normal crossing is ubiquitous in singularity theory and it has a prominent role in the problem of resolution of singularities (see e.g. [1] for the foundations of the classical theory, or [2] for a more recent survey on the field) and in constructing alterations (see e.g. [3]). Informally speaking, finitely many subvarieties of some nonsingular variety W are normal crossing iff at every point of W there exists an “algebraic coordinate system,” such that each of the varieties can be defined locally at the point by coordinate functions. This implies in particular that the varieties themselves cannot have singularities. From this we can obtain the intuition that the normal crossing singularities (that is, points where normal crossing varieties intersect) are, so to say, the best ones after nonsingular points. This is also reflected in the definition of embedded resolution of singularities, which requires that a desingularization morphism π : W → W of an embedded variety X ⊂ W (where W, W are nonsingular) has the property that the irreducible components of the total transform π −1 (X) are normal crossing and the strict transform, i.e. the Zariski closure of π −1 (X \ S), is nonsingular. Here S ⊂ X is the set of singularities of X and π is an isomorphism outside S. As the matter of terminology, π −1 (S) is called the set of exceptional components, which is a union of hypersurfaces in W . In the resolution algorithm of Villamayor (for the theoretical presentation see [4, 5], for the implementation see [6, 7, 8]) it is essential to keep the excepF. Winkler (Ed.): ADG 2002, LNAI 2930, pp. 1–20, 2004. c Springer-Verlag Berlin Heidelberg 2004
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G´ abor Bodn´ ar
tional components normal crossing in the intermediate steps, when the desingularization map π is constructed as the composition of well chosen elementary transformations, called blowups. It is not difficult to choose blowups that create new exceptional components which are normal crossing with the old ones. On the other hand, it is not always possible to have such transformations that at the same time also reduce the multiplicities of the singularities. Therefore the algorithm often has to apply “preparatory” blowups which do not reduce multiplicity; whose only job is to create an advantageous situation, in which the next transformation can achieve
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