Geometry of Rationally Connected Varieties

有理连通变差形成了一类变差，它与曲线和曲面的有理变差和单调变差重合，但在更高的维度上代表了一个非常不同(且表现更好)的类别。研究者研究了有理连通性的一些方面，特别是有理连通性和单圈性之间的关系，以及有理连通性对非代数闭域上簇上的点的存在的影响。关于后者，研究者研究有理连通变量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，来更多地了解多项式方程的解。例如，如果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，代数几何学家引入了簇的一个新性质，称为有理连通性，结果表明多项式方程组的代数与他们定义的簇的有理连通性之间有很强的联系。调查者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.