Volumina der Zariski-Kammern algebraischer Flächen

The big cone on a smooth algebraic surface decomposes naturally into Zariski chambers in a way which is most interesting when considering asymptotic invariants of linear series. This decomposition has so far been studied from a geometric and from a combinatorial point of view. In the present thesis the picture is complemented with a metric perspective by introducing a notion of chamber volume and by giving necessary and sufficient conditions for the volume of a chamber to be finite. Furthermore, methods to inductively calculate chamber volumes are provided by first reducing the calculation of the volume of an arbitrary chamber to the calculation of volumes of nef cones on blow-downs and then giving an explicit formula for the calculation of such nef cone volumes in terms of nef cone volumes of surfaces obtained by blowing down additional exceptional curves. The presented method is used to explicitly calculate chamber volumes on del Pezzo surfaces and on blow-ups of the projective plane whose anticanonical bundle is big.