Explicit resolution of singularities

A resolution of the singularities of an algebraic variety is aparametrization of that variety by a non-singular parameter space.Constructing resolutions is a classical branch in singularity theory with many important applications in geometry, algebra and topology. Recently it has become instrumental for computer aided design with curves and surfaces. Much progress has been made conceptually in the last fourtyyears. However, the effective computation of a resolution is still avery hard problem. We propose to tackle this problem by combining bothhigh-browsed mathematical theory as well as powerful modern computer algebra. Our intended applications are to collect significative experimentation material for the yet unsolved characteristic p case, to develop efficient surface parametrizations for graphic programs, to study canonical classes, and to treat number theoreticproblems related to resolution, e.g. the computation of normal basesof splitting fields. All these objectives are central both theoretically and computationally for the recent research in the field.

Substantial contribution to a better and more explicit understanding of phenomena in resolution of singularities have been made and led to an considerable improvement of algorithmic treatment. For the parametrization problem for rational surfaces and the problem of finding rational points on a Del Pezzo surface, new and effective ideas could successfully be applied. Singular varieties are geometric objects given as the solution set of polynomial equations, which are smooth as soap bubbles with the exception of only a few points. These points, where the variety may intersect itself, form edges or cusps, are called singularities. The resolution of singularities states, that a variety with singularities is the projection of a variety in higher dimension, which has no singularities. In the framework of the project, the original proof by Hironaka has been substantially revised to make it more transparent and provide a more explicit understanding of the occurring phenomena. Subsequently, several improvements on algorithmic resolution of singularities could be derived and implemented. The problems of the still unsolved case of resolution in positive characteristic could be pinned down to some very special and particularly hard cases. Promising new concepts were developed to go about these issues. The new methods and techniques are subject of current research. In algebraic geometry and number theory problems inspired by resolution of singularities have been tackled. For the parametrization problem for rational curves the utilization of techniques from toric geometry and taking advantage of information encoded in the Newton polygon led to two new algorithms. Using a Lie Algebra based approach, an elegant as well as efficient method to find rational points Del Pezzo surfaces has been developed. In addition to the scientific research mentioned above intense examination of possibilities for the visualization of algebraic surfaces has been undertaken. The illustrative material emanating from this engagement represented an important cornerstone in public relations work and found interest and great favour with the general public.