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.

代数几何是研究作为多项式方程组的解集而出现的空间;这样的空间在整个数学和科学中起着重要的作用。代数几何中的一个基本问题是：这样一个空间的几何告诉我们关于决定它的多项式的什么？PI研究的首要目标是使用计算代数中的技术来研究这一主题的开放问题。这项研究将为开源计算代数系统Macaulay 2开发新软件。PI还将继续他的推广活动，以退伍军人在数学奥本大学，与大学的退伍军人资源中心合作。PI将适应技术的几何syzygies从射影几何复曲面几何。特别是，PI将使用交换代数的技术，在Berkesch-Erman-Smith和奥尔洛夫关于环面簇的同调性质的论文上取得进展。PI还将推广，从投影到加权投影设置，著名的绿色定理的线性自由决议的曲线嵌入投影空间。在第三个项目中，PI将开发一个有效的算法，通过推广Eisenbud-Fløystad-Schreyer的算法来计算光滑投射环面簇上的层上同调。PI还将利用Eisenbud-Peeva在此类决议结构方面的工作，解释完全交叉点上自由决议的Fitting理想的周期性现象。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力审查标准进行评估，被认为值得支持。