Birational Geometry and Rational Connectedness

这个项目涉及复数簇的二次几何和有理连通簇的研究。主要目的是研究有理连通簇上有理曲线的空间，目的是构造这些簇的新的双调不变量，并确定这些簇与单调簇之间的关系。第二个目标是进一步发展代数几何的一些技术和工具：发展确定模堆何时有理连接的技术，构造和研究射影空间中光滑有理曲线空间的新紧化，以及改进Kontsevich模堆的Behrend-Manin分层以便它检测多重覆盖。在某些变量集合中的多项式方程组出现在数学、科学和工程的每个分支中。确定这些方程组的解是至关重要的。“一元变分”精确地对应于解的集合最容易描述的方程组--解的集合是具有任意输入的某个多项式序列的所有输出的集合。因此，认识到哪些方程组对应于单旋变量是非常重要的。从几何学的观点来看，一个密切相关的概念是“有理连通性”。在实践中，这个概念要简单得多。潜在地，这两个概念是等价的。如果人们能够证明这两个概念是等价的，这将给出一种更简单的方法来识别哪些方程组对应于单旋变体。调查员打算继续他对这个问题的研究。

This project concerns birational geometry of complex varieties and the study of rationally connected varieties. The main objective is to study spaces of rational curves on rationally connected varieties, with the goal of constructing new birational invariants of these varieties and determining the relationship between these varieties and unirational varieties. The secondary objective is to develop further some of the techniques and tools of algebraic geometry: to develop techniques for determining when moduli stacks are rationally connected, to construct and study new compactifications of the space of smooth rational curves in projective space, and to refine the Behrend-Manin stratification of the Kontsevich moduli stack so that it detects mutliple covers.Systems of polynomial equations in some collection of variables arise in every branch of mathematics, science and engineering. Determining the solutions of these systems of equations is of paramount importance. "Unirational varieties" precisely correspond to those systems of equations where the set of solutions is most easily described -- the set of solutions is the set of all outputs of some sequence of polynomials with arbitrary inputs. So it is very important to recognize which systems of equations correspond to unirational varieties. From the point of view of geometry, a closely related notion is "rational connectedness". This notion is far simpler to recognize in practice. Potentially both notionsare equivalent. If one could prove that both notions areequivalent, this would give a simpler way to recognize which systems of equations correspond to unirational varieties. The investigator intends to continue his research of this problem.