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The Minimal Model Program in Positive and Mixed Characteristics

正特征和混合特征的最小模型程序

基本信息

批准号：
2306854
负责人：
Jakub Witaszek
金额：
$ 16.07万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-10-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306854&HistoricalAwards=false
关键词：
Minimal Model Program Positive Mixed

项目摘要

Algebraic geometry is an active and influential field of mathematics. Its goal is to understand algebraic varieties, which are geometric shapes described by algebraic equations. Not only are such objects essential in many fields of mathematics, but they also come up naturally in other disciplines such as engineering, computer science, or biology. The principal investigator's research centers around the ultimate goal of classifying all algebraic varieties. The PI plans to capitalize on recent breakthroughs in other fundamental fields of mathematics, namely number theory and commutative algebra, to significantly extend the scope of this classification.The principal investigator intends to apply new techniques, related to F-regularity, +-regularity, and mixed characteristic perfections, in order to provide a better understanding of the geometry and commutative algebra of schemes in positive and mixed characteristics. A second aspect of the project is to further the Minimal Model Program in positive and mixed characteristics by focusing on dimensions three and four. The PI plans to apply these results to obtain insight into the liftability of singularities and projective varieties from positive characteristic to characteristic zero, as well as the existence of rational points on varieties over finite fields.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.

代数几何是数学中一个活跃而有影响的领域。它的目标是理解代数簇，这是由代数方程描述的几何形状。这些对象不仅在数学的许多领域是必不可少的，而且在其他学科，如工程，计算机科学或生物学中也很自然地出现。首席研究员的研究中心围绕着分类所有代数簇的最终目标。PI计划利用最近在其他数学基础领域的突破，即数论和交换代数，以显着扩大这一分类的范围。首席研究员打算应用新的技术，与F-正则性，+-正则性，和混合特征完善，为了更好地理解正特征和混合特征概型的几何和交换代数。该项目的第二个方面是通过关注第三和第四个维度，进一步推进积极和混合特征的最小模型方案。 PI计划将这些结果应用于深入了解奇点和投影品种从正特征到零特征的可提升性，以及有限域上品种上合理点的存在性。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。