Birational Geometry and Rational Connectedness

双有理几何和有理关联

• 批准号: 0201423
• 负责人: Aise de Jong
• 金额: $ 6.11万
• 依托单位: Massachusetts Institute of Technology
• 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: 2002-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0201423&HistoricalAwards=false
• 关键词: Birational; Geometry; Rational; Connectedness

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.

该项目涉及复杂簇的双有理几何和有理关联簇的研究。 主要目标是研究有理关联簇上的有理曲线空间，目的是构造这些簇的新双有理不变量并确定这些簇与非有理簇之间的关系。 第二个目标是进一步开发代数几何的一些技术和工具：开发确定模堆栈何时有理连接的技术，构建和研究射影空间中平滑有理曲线空间的新紧化，以及细化 Kontsevich 模堆栈的 Behrend-Manin 分层，以便检测多重覆盖。 数学、科学和工程的每个分支中都会出现一些变量的集合。 确定这些方程组的解至关重要。 “无理簇”精确地对应于那些最容易描述解集的方程组——解集是具有任意输入的某些多项式序列的所有输出的集合。 因此，认识哪些方程组对应于无理簇是非常重要的。 从几何学的角度来看，一个密切相关的概念是“理性连通性”。 这个概念在实践中更容易识别。 可能这两个概念是等效的。 如果能够证明这两个概念是等价的，这将提供一种更简单的方法来识别哪些方程组对应于无理簇。 研究者打算继续研究这个问题。