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Seminar on Algebraic Geometry

Wednesday, 8.3. 14:00-15:00
Seminar room Altenbergerstrasse 50

Hans-Christian Graf v. Bothmer (Hannover)
Counting Components Heuristically.

Via the Weil-Conjectures one can obtain a lot of geometric information about an algebraic variety X defined over ZZ by counting points over a finite field F_p. Unfortunatedly in many interesting examples it is impossible to count all points because there simply are too many of them. By evaluating the equations of X in a number of random points one can estimate the total number of points with sufficient precision to obtain information about the number of components of X and the codimensions of these components. This method is fast and easily parallelizable, but yields results that are only "probably right". In this talk I will explain this method in more detail and show how it can be applied to the construction of rational surfaces in IP^4 and to the Poincaré center problem.