Problems in Algebraic Geometry

拉扎斯菲尔德将工作的一些问题，代数几何。第一系列的问题涉及的代数性质的大程度嵌入的投射品种。定义曲线的方程在过去的25年里得到了深入的研究，但直到最近，人们对更一般的簇的代数行为知之甚少。在这个方向上，拉扎斯菲尔德将继续他的工作与艾因的渐近结构的syzygies高维品种的积极性的嵌入线丛的增长。拉扎斯菲尔德还将致力于一些问题的积极性属性的高余维代数循环。他与德巴雷、艾因和瓦辛的合作产生了这个想法，他的想法是探索在余维1的循环中，积极性的经典概念的更高余维类似物。代数几何是数学中最古老和最核心的领域之一，它研究多元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，从数论到拓扑学、代数和复分析。代数几何在编码理论、理论物理和计算数学等不同领域也有重要的应用。在第一个系列的问题，拉扎斯菲尔德将研究的性质方程切出一个固定的几何轨迹。希望这项研究将开辟新的接触点之间的领域代数几何和交换代数。拉扎斯菲尔德还将研究一些关于较小的几何轨迹可以位于较大的几何轨迹内的方式的问题。

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.