Rationally Connected Varieties

这项建议的主要目的是研究二次几何中的几个问题。第一部分旨在研究有理连通簇，这是从多个角度来看最简单的代数簇。对于非真变种，理性连通性的定义有多种，提出者将研究它们之间的关系。他还将继续研究有理连通族的算术性质。其他部分则集中在簇或对的有界性问题上。特别是，当对数对是对数一般类型时，PI调查者试图建立其体积的一致性质。代数簇粗略地说是由多项式的解描述的图形。代数几何是一门旨在对所有代数簇进行分类的学科。在比生意义上，代数体是由某些特定类型的体构成的。合理关联的品种和一般类型的品种是这样的基本构件。理解它们将在很大程度上提高我们对所有代数变体的理解。

The main aim of this proposal is to investigate a few questions in birational geometry. The first parts intends to study rationally connected varieties, which are the simplest algebraic varieties from many points of view. For non-proper varieties, there are various definitions of rational connectedness.The proposer will study their relations. He will also continue his study on the arithmetic property of rationally connected varieties. The other parts focus on the boundedness type questions of varieties or pairs. Especially, when the pair is of log general type, the PI investigator is trying to establish the uniform properties of their volumes.Algebraic varieties are roughly speaking the figures described by the solutions of polynomials. Algebraic geometry is a subject which aims to classify all algebraic varieties. In the birational sense, algebraic varieties are built up by some specific types of varieties. Rationally connected varieties and general type varieties are such fundamental building blocks. To understand them will largely improve our understanding of all algebraic varieties.