Big cones of algebraic varieties

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Ample line bundles are fundamental objects in modern Algebraic Geometry, which enjoy many geometric, numerical and cohomological properties. By contrast, due to well-known pathologies, line bundles satisfying the more general property of bigness were for a long time considered difficult to treat and geometrically hard to grasp. Very recently, however, there were substantial break-throughs showing that from an asymptotic point of view, big line bundles display quite predictable behaviour analogous to that of ample line bundles -- in this way their geometric use is now much more obvious and they therefore received a great deal of attention. Due to these developments it has become essential to understand the set of big line bundles (the big cone) of an algebraic variety as closely as possible from a structural point of view. In particular, decompositions of the big cone into geometrically and numerically determined subcones are of central importance - they collect line bundles with equivalent geometric behaviour and thus reduce the complexity of the situation drastically. In our group, the following sub-projects in this current area of Algebraic Geometry are to be studied: (A) Big cones of algebraic surfaces, investigation of the chamber numbers and chamber volumes of (in particular) anti-canonical surfaces, geometrical interpretations of the chamber volume and computation of the chamber numbers by algorithmic and combinatorial methods. (B) Big cones of higher dimensional varieties, study of the polyhedral case by methods of the Minimal Model Program, characterization of the sub cones, first treatment of sub cone number and sub cone volume in the non-polyhedral case.