Universal Secant Bundles and Syzygies of Varieties

通用正割束和品种 Syzygies

• 批准号: 2100782
• 负责人: Michael Kemeny
• 金额: $ 16.2万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2100782&HistoricalAwards=false
• 关键词: Universal; Secant; Bundles; Syzygies; Varieties

This research project is concerned with the study of algebraic varieties, in other words geometric spaces that are defined by systems of polynomial equations. The study of the qualitative features of the equations defining algebraic varieties has a long and cherished history in mathematics, beginning with the work of algebraists such as Cayley and Sylvester in the nineteenth century. These questions were put into a modern framework through the pioneering work of Hilbert, who defined a series of invariants known as Betti numbers. The study of these Betti numbers has been an important force in the development of the fields of both algebra and projective geometry for over a hundred years now. These invariants capture the information about the number of equations required to define a variety, as well as the degrees of these equations and the relations amongst them. The aim of this proposal is to study the Betti numbers for several fundamental classes of algebraic varieties. Moreover, the Principal Investigator (PI) will formulate and study more refined conjectures about the rank of the higher relations amongst the defining equations of algebraic varieties. This provides more detail into the structure of these equations than can be provided by the Betti numbers alone. During this award, the PI will study the Betti numbers of several classes of algebraic varieties using new techniques such as the technique of Universal Secant Bundles. The PI has previously applied this technique successfully on a series of questions about the Betti numbers of algebraic curves which were first asked in the 1980s in the work of mathematicians such as Green and Lazarsfeld. The PI will study these invariants in new settings, such as the case of higher dimensional varieties, with a particular focus on understanding the equations and syzygies of Veronese varieties and Abelian surfaces. Moreover, the PI will provide special, explicit bases consisting of syzygies of minimal possible rank for the syzygy spaces of several classes of algebraic varieties, generalizing well-known work on Green on the generation of the ideal of canonical curves by quadrics of rank four. More generally, the PI hopes to formulate new conjectures and questions which will open up new lines of inquiry for the field as a whole.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该研究项目与代数品种的研究有关，换句话说，几何空间是由多项式方程式定义的。对定义代数品种的方程式的定性特征的研究在数学方面具有悠久而珍贵的历史，从19世纪的代数主义者（例如Cayley和Sylvester）的工作开始。这些问题是通过希尔伯特（Hilbert）的开创性工作将现代框架置于现代框架中的，希尔伯特（Hilbert）定义了一系列被称为贝蒂（Betti）数字的不变性。一百多年来，对这些Betti数量的研究一直是代数和投射几何形状田地发展的重要力量。这些不变的人捕获了有关定义一个方程所需的方程数以及这些方程式以及它们之间关系的信息。该提案的目的是研究几种代数品种基本类别的贝蒂数字。此外，首席研究员（PI）将制定和研究有关代数品种定义方程之间较高关系等级的更精致的猜想。与仅贝蒂数字所能提供的相比，这提供了这些方程结构的详细信息。在此奖励期间，PI将使用新技术（例如通用叶线捆绑包的技术）来研究几类代数品种的Betti数量。 PI此前曾成功地应用了有关贝蒂（Betti）数量的代数曲线的一系列问题，这些曲线在1980年代首次在数学家（例如Green和Lazarsfeld）的工作中提出。 PI将在新环境中研究这些不变性，例如较高的尺寸品种的情况，特别着眼于了解Veronese品种和Abelian表面的方程式和共同点。此外，PI将提供特殊的，明确的基础，包括几类代数品种的Syzygy空间的最小排名，将绿色的著名工作推广到第四级四边形。更笼统地，PI希望制定新的猜想和问题，这些猜想和问题将为整个领域开辟新的询问线。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估标准来通过评估来支持的。