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.

该研究项目涉及代数簇的研究，即由多项式方程组定义的几何空间。对定义代数簇的方程的定性特征的研究在数学领域有着悠久而珍贵的历史，始于十九世纪凯莱和西尔维斯特等代数学家的工作。通过希尔伯特的开创性工作，这些问题被纳入现代框架中，他定义了一系列称为贝蒂数的不变量。一百多年来，对这些贝蒂数的研究一直是代数和射影几何领域发展的重要力量。这些不变量捕获有关定义变量所需的方程数量以及这些方程的次数以及它们之间的关系的信息。本提案的目的是研究几个基本类代数簇的 Betti 数。此外，首席研究员（PI）将制定和研究关于代数簇定义方程之间的高级关系的等级的更精细的猜想。与单独的贝蒂数相比，这为这些方程的结构提供了更多细节。在这个奖项期间，PI 将使用通用正割束技术等新技术来研究几类代数簇的 Betti 数。 PI 此前已成功地将这种技术应用于一系列有关代数曲线 Betti 数的问题，这些问题是在 20 世纪 80 年代 Green 和 Lazarsfeld 等数学家的工作中首次提出的。 PI 将在新的环境中研究这些不变量，例如高维簇的情况，特别关注理解维罗内簇和阿贝尔曲面的方程和 syzygies。此外，PI 将为几类代数簇的 syzygy 空间提供特殊的、显式的基，由最小可能秩的 syzygies 组成，概括 Green 中关于通过四阶二次曲线生成理想正则曲线的著名工作。更广泛地说，PI 希望提出新的猜想和问题，从而为整个领域开辟新的探究路线。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响进行评估，被认为值得支持审查标准。