Higher rational connectedness and applications

更高的理性连接和应用

• 批准号: 0758521
• 负责人: Jason Starr
• 金额: $ 9.79万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0758521&HistoricalAwards=false
• 关键词: Higher; rational; connectedness; applications

Rational simple connectedness is an algebraic notion which is to simple connectedness as rational connectedness is to path connectedness.Just as a topological fibration over a 2-dimensional base with simply connected fiber admits a continuous section, also an algebraic fibration over a surface with rationally simply connected general fiber admits a rational section (under suitable additional hypotheses). This project investigates the theory beyond this result, just as topological obstruction theory is the theory beyond the quoted topological result. The first goal is to determine how the obstruction to "weak approximation"(approximation of power series solutions by polynomial solutions) decomposes into a local obstruction and a global obstruction. The second goal is to investigate the obstruction theory where it is not yet known by determining precisely which algebraic fibrations over a surface of a specified, simple type admit a rational section.Systems of polynomial equations are ubiquitous in mathematics, science and engineering. In studying the collection of all solutions in complex numbers, i.e., the variety, associated to such a system, there is one special phenomenon: the system is "rationally connected" if for every pair of solutions, there is a polynomial map taking values in the variety and whose values interpolate between the given pair of solutions. This special property is often satisfied in practice. Surprisingly, a system of polynomial equations depending algebraically on 1 extra parameter (often thought of as time) always has a family of solutions varying as a polynomial of the parameter so long as the system for a fixed general choice of the parameter is rationally connected. There is now an analogous theorem for a 2-parameter system, but with very strong constraints on the system. The goal of the project is to weaken the constraint condition, and thus make the advance more widely applicable, by using notions analogous to those in topology, i.e., "rubber-sheet geometry".

有理简单连通性是一个代数概念，它对于简单连通性就像有理连通性对于路径连通性一样。正如具有简单连接纤维的二维基底上的拓扑纤维允许连续截面一样，具有有理简单连通性的表面上的代数纤维也是如此连接的普通光纤允许有理部分（在适当的附加假设下）。 该项目研究了超出该结果的理论，正如拓扑阻碍理论是超出所引用的拓扑结果的理论一样。第一个目标是确定“弱近似”（通过多项式解逼近幂级数解）的障碍如何分解为局部障碍和全局障碍。 第二个目标是通过精确确定特定简单类型表面上的哪些代数纤维允许有理截面来研究尚不清楚的阻碍理论。多项式方程组在数学、科学和工程中无处不在。 在研究复数中所有解的集合（即与此类系统相关的多样性）时，存在一个特殊现象：如果对于每一对解，存在一个取值的多项式映射，则该系统是“有理连接的”多样性及其值在给定的一对解之间进行插值。 这种特殊性质在实践中常常得到满足。 令人惊讶的是，只要参数的固定一般选择的系统是有理连接的，代数上依赖于 1 个额外参数（通常被认为是时间）的多项式方程组总是具有一系列作为参数多项式变化的解。 现在对于 2 参数系统有一个类似的定理，但对系统有非常强的约束。 该项目的目标是通过使用类似于拓扑学的概念，即“橡胶板几何”，削弱约束条件，从而使这一进展得到更广泛的应用。