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.

拉扎斯菲尔德将致力于解决代数几何中的许多问题。第一系列问题涉及射影簇的大次嵌入的代数性质。在过去的二十五年里，人们对定义曲线的方程进行了深入的研究，但直到最近，人们对更一般簇的代数行为知之甚少。在这个方向上，随着嵌入线丛正性的增长，Lazarsfeld 将继续与 Ein 一起研究高维簇齐性的渐近结构。拉扎斯菲尔德还将研究一些有关高维代数循环的正性质的问题。源于他与 Debarre、Ein 和 Voisin 的合作，这里的想法是探索余维一循环的经典正性概念的更高余维类似物。代数几何是最古老和最核心的数学领域之一，研究多变量多项式方程组解的几何性质。它与许多其他数学领域建立了联系，从数论到拓扑、代数和复分析。代数几何还在编码理论、理论物理和计算数学等不同领域的问题中找到了重要的应用。在第一系列问题中，拉扎斯菲尔德将研究切出固定几何轨迹的方程的性质。希望这项研究能够在代数几何和交换代数领域之间开辟新的接触点。拉扎斯菲尔德还将研究一些关于较小几何基因座如何位于较大几何基因座内的猜想。