Multigraded commutative algebra and the geometry of syzygies

多级交换代数和 syzygies 几何

• 批准号: 2302373
• 负责人: Michael Brown
• 金额: $ 22万
• 依托单位: Auburn University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302373&HistoricalAwards=false
• 关键词: Multigraded; commutative; algebra; geometry; syzygies

Algebraic geometry is the study of spaces that arise as solution sets to systems of polynomial equations; such spaces play an important role throughout mathematics and the sciences. A fundamental question in algebraic geometry is: what does the geometry of such a space tell one about the polynomials that determine it? The overarching goal of the PI’s research is to use techniques in computational algebra to study open problems on this theme. This research will lead to the development of new software for the open-source computational algebra system Macaulay2. The PI will also continue his outreach to veterans in mathematics at Auburn University, in collaboration with the university’s Veterans Resource Center.The PI will adapt the techniques of the geometry of syzygies from projective geometry to toric geometry. In particular, the PI will use techniques in commutative algebra to make progress on conjectures of Berkesch-Erman-Smith and Orlov on the homological properties of toric varieties. The PI will also generalize, from the projective to the weighted projective setting, a celebrated theorem of Green on the linearity of free resolutions of curves embedded in projective space. In a third project, the PI will develop an efficient algorithm for computing sheaf cohomology over smooth projective toric varieties by generalizing an algorithm due to Eisenbud-Fløystad-Schreyer that applies to sheaves on projective space. The PI will also explain a periodicity phenomenon for the Fitting ideals of free resolutions over complete intersections, leveraging work of Eisenbud-Peeva on the structure of such resolutions.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.

代数几何是对作为多项式方程组的解集而出现的空间的研究；这种空间在数学和科学中发挥着重要作用。代数几何的一个基本问题是：这种空间的几何告诉我们什么。 PI 研究的首要目标是使用计算代数技术来研究该主题的开放问题。这项研究将导致开源计算代数系统的新软件的开发。麦考利2. PI 还将与奥本大学的退伍军人资源中心合作，继续向数学领域的退伍军人进行推广。PI 会将 syzygies 几何技术从射影几何改为环面几何。使用交换代数技术在 Berkesch-Erman-Smith 和 Orlov 关于环面簇同调性质的猜想上取得进展 PI 也将从射影进行推广。在第三个项目中，PI 将开发一种有效的算法，通过推广一种算法来计算平滑射影环面簇上的束上同调。适用于射影空间上滑轮的 Eisenbud-Fløystad-Schreyer 还将解释完全交叉点上自由分辨率拟合理想的周期性现象，利用 Eisenbud-Peeva 在此类决议结构上的工作。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。