Big cones of algebraic varieties

充填线丛是现代代数几何中的基本对象，它具有许多几何、数值和上同调性质。相比之下，由于众所周知的病理，满足更普遍的大的属性的线束在很长一段时间内被认为很难治疗，几何上很难把握。然而，最近有大量的突破表明，从渐近的观点来看，大的线束表现出与充足的线束类似的非常可预测的行为--这样，它们的几何用途现在更加明显，因此它们受到了极大的关注。由于这些发展，从结构的角度尽可能接近地理解代数簇的大线丛(大锥体)的集合变得至关重要。特别是，将大圆锥体分解成几何和数字确定的子圆锥体至关重要--它们收集具有相同几何行为的线束，从而极大地降低了情况的复杂性。在本课题组中，将研究当前代数几何领域的以下子项目：(A)代数曲面的大圆锥，(特别是)反正则曲面的腔数和腔体积的调查，腔体积的几何解释以及用算法和组合方法计算腔数。(B)高维变化的大锥体，用最小模型程序的方法研究多面体情况，亚锥体的表征，非多面体情况下的子锥体数目和亚锥体体积的第一处理。

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.