Rationally Connected Varieties

本文的主要目的是研究双有理几何中的几个问题。第一部分研究了有理连通簇，从多个角度来看，有理连通簇是最简单的代数簇。对于非真簇，有理连通性有各种不同的定义，提出者将研究它们之间的关系。他还将继续他的研究算术财产的合理连接品种。另一部分主要讨论簇或对的有界性问题。特别是，当对是日志一般类型，PI调查员试图建立统一的性质，他们的体积。代数品种粗略地说，由多项式的解决方案所描述的数字。代数几何是一门旨在对所有代数簇进行分类的学科。在双有理意义下，代数簇是由某些特定类型的簇构成的。理性连接的变种和一般类型变种就是这样的基本构建块。理解它们将在很大程度上提高我们对所有代数簇的理解。

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.