Topics in Singularity Theory and Commutative Algebra

奇点理论和交换代数专题

• 批准号: 0754208
• 负责人: Steven Cutkosky
• 金额: $ 20.7万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2013-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0754208&HistoricalAwards=false
• 关键词: Topics; Singularity; Theory; Commutative; Algebra

Cutkosky will investigate four projects: toroidalization, local monomialization and ramification of surfaces in positive characteristic, asymptotic properties of powers of ideals and semigroups of valuations on local rings. In toroidalization, the main conjecture is to show that any dominant morphismof varieties can be made into a toroidal mapping, or a local monomial mapping, by blowing up nonsingular subvarieties in the domain and range. Cutkosky will extend his recent work proving the conjecture for 3-folds,towards a general proof of the conjecture. On the problem of ramification in positive characteristic,the proposer has found with Oliver Piltant stable forms of mappings between germs of characteristic p surfaces.Cutkosky proposes to refine the formulation of these forms, and apply them to a study of ramification of maps of surfaces in characteristic p. Cutkosky will continue his work on computing asymptotic properties of Hilbert type functions on functors of powers of ideals. Cutkosky also seeks to understand the semigroups which canoccur as the value semigroup of a (possible noetherian) valuation dominating a noetherian local domain.Algebra is of increasing importance in modern science and technology. Cutkosky will investigateseveral problems of theoretical importance in the general area of Commutative Algebra and Algebraic Geometry.The main objective is to understand the structure of polynomial mappings, and systems of polynomial equations. Cutkosky will train undergraduate and graduate students, and mentor young researchers through this project.

Cutkosky将研究四个项目：toroidalization，局部单单性化和分歧的表面在积极的特征，渐近性质的权力的理想和半群的估值局部环。 在环面化中，主要的猜想是证明任何簇的支配态射都可以通过爆破定义域和值域中的非奇异子簇而成为一个环面映射或一个局部单项式映射。Cutkosky将扩展他最近的工作证明猜想的3倍，对一个一般的证明猜想。在正特征的分歧问题上，提出者与奥利弗一起发现了特征p曲面芽之间映射的稳定形式。Cutkosky提出改进这些形式的公式，并将其应用于特征p曲面映射的分歧研究。Cutkosky将继续他的工作，计算Hilbert型函数关于理想幂的函子的渐近性质。Cutkosky还试图理解半群，它可以作为一个（可能的Noether）赋值控制一个Noether局部区域的值半群出现。代数在现代科学和技术中越来越重要。Cutkosky将研究交换代数和代数几何一般领域中的几个理论重要性问题。主要目标是了解多项式映射的结构和多项式方程组。卡特科斯基会 通过这个项目培训本科生和研究生，并指导年轻的研究人员。