Geometry of Rationally Connected Varieties

理性连接的变种形成了一类变种，与曲线和曲面的理性和非理性变种相一致，但在高维中代表了一个非常不同（并且表现更好）的类别。研究者研究理性连通性的各个方面；特别地，在非代数闭域上讨论了有理连通性与唯一性的关系，以及有理连通性对点的存在性的影响。在后者方面，研究者研究了有理曲线在理性连接变项上的参数空间几何。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（称为变量）来更多地了解多项式方程的解。例如，如果一个椭圆是由两个变量的多项式给出的，我们可能会问椭圆的形状如何影响多项式的解。最近，代数几何学者引入了一种新的变种的性质，称为有理连通性，迄今为止的结果表明多项式方程系统的代数与它们所定义的变种的有理连通性之间有很强的联系。调查员进一步研究了这种联系。

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.