Commutative Algebra and Algebraic Geometry

交换代数和代数几何

• 批准号: 1502190
• 负责人: David Eisenbud
• 金额: $ 53.19万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1502190&HistoricalAwards=false
• 关键词: Commutative; Algebra; Algebraic; Geometry

Algebraic geometry is the study of the qualitative properties of forms described by polynomial equations. The questions it addresses arise in many areas of science and mathematics, from statistical sampling to robotics and mathematical physics. This research project concerns the study of aspects of algebraic geometry related to syzygy theory, a sort of microscope that allows one to see details of polynomial equations that are not apparent from the equations themselves.The investigator will work on problems in commutative algebra, algebraic geometry and computational methods for these fields. He will focus on four areas: 1) Cohen-Macaulay and Ulrich complexity of hypersurfaces: A central problem is to describe the minimal size of a non-free maximal Cohen-Macaulay module over a hypersurface, in particular over the permanent hypersurfaces. 2) The structure of high syzygies over complete intersections: Central problems are to give criteria for a module over a complete intersection to be a "high syzygy" in the sense of Eisenbud and Peeva; and to understand the relations between the even and odd parts of the infinite minimal free resolutions of modules over a complete intersection. 3) Duality for residual intersections: The central remaining open problem is to prove and extend a conjecture of van Straten and Warmt on the socle of a particular representation of the canonical module of a 0-dimensional residual intersection. 4) Tate resolutions for complexes of sheaves on a toric variety: The central problem is to define and compute a good analogue for toric varieties of the Tate resolution of a sheaf or a complex of sheaves on a projective space.

代数几何是研究由多项式方程描述的形式的定性性质。它解决的问题出现在科学和数学的许多领域，从统计抽样到机器人和数学物理。本研究课题是关于与合冲理论相关的代数几何的研究。合冲理论是一种显微镜，可以看到多项式方程本身不明显的细节。研究者将研究交换代数、代数几何和这些领域的计算方法中的问题。他将专注于四个领域：1）超曲面的Cohen-Macaulay和Ulrich复杂性：一个中心问题是描述超曲面上的非自由极大Cohen-Macaulay模的最小尺寸，特别是在永久超曲面上。 2)完全交上的高合合的结构：中心问题是给出完全交上的模是Eisenbud和Peeva意义下的“高合合”的判据;并理解完全交上的模的无限极小自由分解的偶数部分和奇数部分之间的关系。3)剩余交集的对偶性：中心剩余开放问题是证明和扩展货车Straten和Warmt关于0维剩余交集的典范模的特定表示的基座的猜想。 4)复曲面簇上的层复形的泰特分解：中心问题是定义和计算射影空间上的层或层复形的泰特分解的复曲面簇的一个好的类似物。