Geometry of Rationally Connected Varieties

有理连接品种的几何

• 批准号: 0200659
• 负责人: Joseph Harris
• 金额: $ 22.5万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2005-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200659&HistoricalAwards=false
• 关键词: Geometry; Rationally; Connected; Varieties

Rationally connected varieties form a class of varieties that coincides with rational and unirational varieties for curves and surfaces, but represents a very different (and better behaved) class in higher dimensions. The investigator studies aspects of rational connectivity; in particular, the relation between rational connectivity and unirationality and the implications of rational connectivity for existence of points on varieties over non-algebraically closed fields. In relation to the latter, the investigator studies the geometry of parameter spaces for rational curves on a rationally connected variety.The central goal of algebraic geometry is to learn more about the solutions of polynomial equations by studying geometric objects, called varieties, associated to them. For example, if an ellipse is given to us by a polynomial in two variables, we might ask how the shape of an ellipse affects the solutions of the polynomial. Recently, algebraic geometers have introduced a new property of varieties, called rational connectivity, and results to far indicate a strong connection between the algebra of a system of polynomial equations and the rational connectivity of the variety they define. The investigator studies this connection further.

有理连通的簇形成了一类簇，与曲线和曲面的有理和单有理簇一致，但在更高的维度上代表了一个非常不同的（并且表现更好的）类别。调查研究方面的合理连接，特别是合理的连接和unirationality之间的关系和合理的连接的影响存在的点品种在非代数闭域。关于后者，研究者研究有理连通簇上有理曲线的参数空间的几何。代数几何的中心目标是通过研究与多项式方程相关的几何对象（称为簇）来了解更多关于多项式方程的解。例如，如果一个椭圆是由一个二元多项式给出的，我们可能会问椭圆的形状如何影响多项式的解。最近，代数几何学家引入了一个新的性质的品种，称为合理的连接，结果远远表明之间的强连接代数系统的多项式方程和合理的连接品种，他们定义。研究人员进一步研究了这种联系。