CAREER: Higher rational connectedness, higher Fano manifolds, and applications

职业：更高的理性连通性、更高的 Fano 流形和应用

• 批准号: 0846972
• 负责人: Jason Starr
• 金额: $ 42万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0846972&HistoricalAwards=false
• 关键词: CAREER; Higher; rational; connectedness; higher

Rational connectedness, rational simple connectedness, etc., are algebro-geometric analogues of path connectedness, simple connectedness, etc. Just as higher connectedness and higher homotopy groups play an important role in topological obstruction theory, so the algebro-geometric analogues should play an important role in algebro-geometric obstruction theory. In particular, new ideas coming from A1-homotopy theory should resolve the weak approximation problem of Hassett and Tschinkel. There are 3 objectives of the research component. One objective is to develop an algebro-geometric theory of higher rational connectedness analogous to the theory of topological obstruction theory, and with applications to existence of rational sections ofalgebraic fibrations. A second objective is the classification of higher Fano manifolds, the study of their connection to higher rational connectedness, and the study of their complex differential geometry. The final objective is to study the parameter spaces for rational curves on Fano manifolds which do not satisfy the higher Fano conditions. This is relevant to the open problem of proving existence of non-unirational Fano manifolds.Quite frequently in science and engineering one wants to solve a system of polynomial equations in some set of variables, and depending on some set of parameters. An optimal case is when there is a solution to the system of equations whose coordinates are themselves polynomials (or more often fractions of polynomials) in the parameters. There are some sufficient conditions for this optimal solution which involve the geometry of the solution set for a general choice of the parameters. The goal is to sharpen these results to give conditions which are both sufficient and necessary, i.e., to develop a theory of "obstructions" to the existence of rational solutions.The educational component of the proposal has 3 parts: a program aimed at training high school math teachers from the MA program directed by the PI so that they may establish and run math clubs and math circles in their schools, a seminar/summer workshop in mathematical exposition for graduate students and recent postdocs, and a northeastern regional algebraic geometry seminar fostering interactions between graduate students in different cities across the northeastern United States.

理性连通性，理性简单连通性，等等，是路连通性、单连通性等的代数几何类比。正如高连通性和高同伦群在拓扑阻塞理论中起着重要作用一样，代数几何类比在代数几何阻塞理论中也应该起着重要作用。 特别是，来自A1-同伦理论的新思想应该解决Hassett和Tschinkel的弱近似问题。研究部分有三个目标。一个目标是发展一个代数几何理论的更高的合理的连通性类似的理论的拓扑阻塞理论，并与应用程序的存在合理的部分ofalgebraic纤维化。第二个目标是分类较高的法诺流形，研究他们的连接，以更高的理性连通性，并研究其复杂的微分几何。最终目的是研究Fano流形上不满足高阶Fano条件的有理曲线的参数空间。这与证明非单有理Fano流形的存在性的公开问题有关。在科学和工程中，人们经常希望解决某组变量的多项式方程组，并取决于某组参数。 一个最佳的情况是当存在一个方程组的解时，该方程组的坐标本身是参数中的多项式（或更常见的是多项式的分数）。 有一些充分条件，这个最佳解决方案，其中涉及几何的解决方案集的一般选择的参数。 我们的目标是锐化这些结果，以给出既充分又必要的条件，即，发展一种“障碍”理论，以存在合理的解决办法。建议的教育部分有三个部分：一个旨在培训由PI指导的MA课程的高中数学教师的计划，以便他们可以在学校建立和运行数学俱乐部和数学圈，为研究生和最近的博士后举办数学博览会的研讨会/夏季讲习班，以及东北地区代数几何研讨会，促进美国东北部不同城市研究生之间的互动。