Problems in Algebraic Geometry

代数几何问题

• 批准号: 1439285
• 负责人: Robert Lazarsfeld
• 金额: $ 53.22万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1439285&HistoricalAwards=false
• 关键词: Problems; Algebraic; Geometry

Lazarsfeld will work on a number of problems in algebraic geometry. A first series of questions concerns the algebraic properties of large degree embeddings of projective varieties. The equations defining curves have been studied intensively over the last twenty-five years, but until recently little was known about the algebraic behavior of more general varieties. In this direction, Lazarsfeld will continue his work with Ein on the asymptotic structure of the syzygies of higher-dimensional varieties as the positivity of the embedding line bundle grows. Lazarsfeld will also work on a number of questions concerning the positivity properties of higher-codimension algebraic cycles. Growing out of his work with Debarre, Ein and Voisin, the idea here is to explore the higher-codimension analogues of classical notions of positivity for cycles of codimension one. Algebraic geometry, one of the oldest and most central fields of mathematics, studies geometric properties of solutions to systems of polynomial equations in several variables. It makes connections with many other fields of mathematics, ranging from number theory to topology, algebra, and complex analysis. Algebraic geometry has also found important applications to problems in such diverse areas as coding theory, theoretical physics and the mathematics of computation. In a first series of problems, Lazarsfeld will study the properties of the equations cutting out a fixed geometric locus. It is hoped that this research will open up new points of contact between the fields of algebraic geometry and commutative algebra. Lazarsfeld will also work on some conjectures concerning the manner in which smaller geometric loci can sit inside larger ones.

Lazarsfeld将在代数几何形状中处理许多问题。第一个问题涉及大型射影品种嵌入的代数特性。在过去的二十五年中，对定义曲线的方程式进行了深入研究，但直到最近，对更一般品种的代数行为知之甚少。在这个方向上，Lazarsfeld将继续与EIN一起在较高维度品种的Syzygies的渐近结构上进行工作，这是嵌入线束的阳性增长。 Lazarsfeld还将在许多有关高统计代数代数周期的阳性特性的问题上工作。从他与Debarre，Ein和Voisin的工作中成长出来，这里的想法是探索对编码循环的经典阳性概念的更高补充类似物。代数几何形状是数学的最古老，最中心的领域之一，研究了几种变量中多项式方程系统的几何特性。它与许多其他数学领域建立了联系，从数字理论到拓扑，代数和复杂分析。代数几何形状还发现了在编码理论，理论物理学和计算数学等各个领域的问题中的重要应用。在第一个系列问题中，Lazarsfeld将研究切除固定几何基因座的方程式的性能。希望这项研究能够为代数几何和交换代数的领域之间的新接触点开辟新的接触点。拉扎尔斯菲尔德（Lazarsfeld）还将对一些猜想进行，以了解较小的几何基因座可以坐在较大的基因座的方式。