Computational Formal Resolution of Surfaces in Pc

In Algebraic Geometry smooth varieties are (in general) well-understood. By contrast (or simply because of that) the objects of interest are often singular varieties. From the theoretical point of view a remedy for this situation is the famous Theorem of Hironaka [8] on the resolution of singularities: If X is a variety over a field of characteristic 0 then there always exists a smooth variety X̃ and a proper birational morphism π : X̃ → X. So for proving theorems and defining birational invariants, one can argue on X̃ rather than on X and finally transfer the result back to the singular variety. This theorem has been made constructive by Villamayor [10], Bierstone-Milman [3] and others. There are also two implementations of the resolution algorithm in Singular, one by Anne Frübis-Krüger [6] and another by Gábor Bodnár and the second author [4]. In principal this makes many theoretical results algorithmic, but any algorithm relying on a resolution suffers from the high computational complexity of the resolution process. From the computational point of view it is not always necessary to describe the resolution completely. In the case of algebraic curves over a field K series expansion have proved to be an important algorithmic tool. Here the preimage of the singular locus w.r.t. a resolution of singularities is a finite set of points. The idea is to describe the resolution by power series expansions that determine analytic (or formal) neighborhoods of these points. If K has characteristic 0 Puiseux expansions can be used and the Newton-Puiseux algorithm is implemented in many computer algebra systems including Magma, Maple and Singular. The latter system also contains an implementation of Hamburger-Noether expansions [5], that provide a similar tool in positive characteristic. Applications are for example the computation of integral bases in the function field [9], the computation of linear systems of divisors [7] and as a special case systems of adjoint curves used for parametrization [2]. To our knowledge for higher dimensional varieties there is no similar tool (at least no accessible implementation). The purpose of this talk is to present such a tool for surfaces in PC and its implementation in Magma. This is achieved by the following means. Date: June 30, 2006. The authors were supported by the FWF (Austrian Science Fund) in the frame of the research projects SFB 1303 and P15551.