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".

理性简单连接是一个代数概念，它是简单的连接，因为理性连接是与路径连接性的。作为二维基础上的拓扑振动，具有简单连接的纤维的拓扑振动，可以接收连续的部分，在表面上也可以简单地连接的一般纤维在合理的一般纤维上（在合适的一般纤维中）（在表面上）。 该项目研究了超出这一结果的理论，就像拓扑阻塞理论是超出引用拓扑结果的理论一样。第一个目标是确定“弱近似”（通过多项式溶液对功率序列溶液的近似）的阻塞分解为局部阻塞和全球障碍。 第二个目标是通过精确确定在指定的，简单类型的表面上确定哪​​些代数振动来研究障碍理论，该代数纤维在数学，科学和工程中无处不在。 在研究所有解决方案的收集，即与该系统相关的品种时，有一个特殊的现象：如果对于每对溶液，系统都是“合理连接的”，则有一个多项式映射在多种方面的值，并且其值在给定的解决方案之间插值。 在实践中，这种特殊属性通常会得到满足。 出乎意料的是，只要固定的一般选择参数的系统与理性选择的系统，始终是参数的多项式变化的多项式方程系统（通常被认为是时间的）始终具有一个溶液家族的变化。 现在有一个针对2参数系统的类似定理，但是系统对系统有很强的限制。 该项目的目的是通过使用类似于拓扑的概念，即“橡皮图几何形状”，从而削弱约束条件，从而使进步更广泛地适用。