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Big cones of algebraic varieties

代数簇的大锥体

基本信息

批准号：
229842420
负责人：
Professor Dr. Thomas Bauer
金额：
--
依托单位：
Arbeitsgruppe Mathematik und ihre Didaktik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2012
资助国家：
德国
起止时间：
2011-12-31 至 2014-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/229842420?language=en
关键词：
Big cones algebraic varieties

项目摘要

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.

充足线丛是现代代数几何中的基本对象，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，满足更一般的大属性的线束在很长一段时间内被认为是难以处理和几何难以把握。然而，最近，有大量的break-dampers显示，从渐近的角度来看，大线束显示相当可预测的行为类似于丰富的线束-以这种方式，他们的几何用途现在是更加明显，因此他们收到了很大的关注。由于这些发展，从结构的角度尽可能地理解代数簇的大线丛（大锥）的集合变得至关重要。特别是，分解的大锥成几何和数值确定的子锥是至关重要的-他们收集线束与等效的几何行为，从而大大降低了复杂的情况。在我们的小组中，以下子项目在当前的代数几何领域进行研究：（A）代数曲面的大圆锥，调查室数和室体积（特别是）反正则曲面，几何解释室体积和计算室数的算法和组合方法。(B)大锥的高维品种，研究的多面体情况下的最小模型程序的方法，表征的子锥，第一次治疗的子锥数量和子锥体积在非多面体的情况下。