Universal Secant Bundles and Syzygies of Varieties

这项研究项目涉及代数簇的研究，换句话说，即由多项式方程组定义的几何空间。定义代数簇的方程的定性特征的研究在数学上有着悠久的历史，始于19世纪的代数学家凯利和西尔维斯特的工作。通过希尔伯特的开创性工作，这些问题被放入了一个现代框架中，他定义了一系列被称为贝蒂数的不变量。一百多年来，对这些Betti数的研究一直是代数和射影几何领域发展的重要力量。这些不变量捕捉有关定义变化所需的方程式数量的信息，以及这些方程式的阶数和它们之间的关系。这一建议的目的是研究几类基本代数簇的Betti数。此外，首席调查员(PI)将制定和研究关于代数变种定义方程之间更高关系的等级的更精细的猜想。这为这些方程的结构提供了比仅由Betti数提供的更详细的信息。在这个奖项中，PI将使用新的技术，如万能割丛技术，研究几类代数簇的Betti数。PI以前曾成功地将这一技术应用于一系列关于代数曲线的Betti数的问题，这些问题是在20世纪80年代格林和拉扎斯菲尔德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Verones簇和Abel曲面的方程和合子。此外，PI将为几类代数簇的合集空间提供由最小可能秩合组成的特殊的显式基，推广了Green关于用四阶二次曲线生成标准曲线理想的著名工作。更广泛地说，PI希望制定新的猜测和问题，为整个领域开辟新的调查路线。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.