Universal Secant Bundles and Syzygies of Varieties

这个研究项目涉及代数簇的研究，换句话说，是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中有着悠久而珍贵的历史，始于世纪的凯莱和西尔维斯特等代数学家的工作。这些问题通过希尔伯特的开创性工作被纳入现代框架，他定义了一系列被称为贝蒂数的不变量。研究这些贝蒂数一直是一个重要的力量，在发展领域的代数和射影几何超过一百年了。这些不变量捕获了关于定义一个变量所需的方程数量的信息，以及这些方程的阶数和它们之间的关系。这个建议的目的是研究几个基本类的代数簇的贝蒂数。此外，首席研究员（PI）将制定和研究更精细的代数簇的定义方程之间的高级关系的排名。这提供了更多的细节到这些方程的结构比可以提供的贝蒂数单独。在这个奖项，PI将研究贝蒂数的几类代数簇使用新的技术，如技术的通用割线丛。PI以前曾成功地应用这种技术对一系列问题的贝蒂数的代数曲线，这是第一次问在20世纪80年代的工作中的数学家，如绿色和Lazarsfeld。PI将在新的设置中研究这些不变量，例如高维簇的情况，特别关注理解Veronese簇和Abel曲面的方程和syzygies。此外，PI将提供特殊的，明确的基础组成的syzygies的最小可能的秩的syzygies空间的几个类的代数簇，推广著名的工作绿色的代理想的规范曲线的秩四。更广泛地说，PI希望制定新的计划和问题，为整个领域开辟新的调查路线。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.