Problems in Algebraic Geometry

拉扎斯菲尔德将工作的一些问题，在复杂的代数几何。首先，与穆斯塔，他将详细调查的程序，其中一个关联凸体的线性系列的投影品种。介绍了在路过的Okounkov，这种建设有可能揭示重要的新的光的结构的线性系统。第二组问题涉及使用乘数理想和相关工具从高维几何研究一些问题的本质上是本地性质。具体而言，拉扎斯菲尔德希望解决一些programmures德费内克斯，艾因和穆斯塔塔有关代数不变量的理想，并证明在所有维度的结果衰减的奇异性分级家庭的理想所建议的一个定理的法弗尔-琼森有关电流的飞机。代数几何是数学中最古老和最核心的领域之一，它涉及多项式方程组解的几何研究。它涉及许多其他数学领域，从数论到拓扑学，代数和复分析。它已经发现了重要的应用问题，在这样不同的领域，如编码理论和理论物理。特别是问题，拉扎斯菲尔德将研究涉及有关问题的代数几何性质的固体在空间和特殊集合的多项式。希望这项工作将导致在该领域的一些有价值的新技术的发展。

Lazarsfeld will work on a number of problems in complex algebraic geometry. First, with Mustata, he will investigate in detail a procedure by which one associates a convex body to a linear series on a projective variety. Introduced in passing by Okounkov, this construction has the potential to shed important new light on the structure of linear systems. A second set of problems involves using multiplier ideals and related tools from higher dimensional geometry to study some questions of an essentially local nature. Specifically, Lazarsfeld hopes to resolve some conjectures of De Fernex, Ein and Mustata relating algebraic invariants of an ideal, and to prove in all dimensions a result about attenuation of singularities of graded families of ideals suggested by a theorem of Favre-Jonsson concerning currents in the plane. Algebraic geometry, one of the oldest and most central fields of mathematics, deals with the geometric study of the solutions of systems of polynomial equations. It touches on many other fields of mathematics, ranging from number theory to topology, algebra and complex analysis. It has found important applications to problems in such diverse areas as coding theory and theoretical physics. The particular questions that Lazarsfeld will study involve relating questions in algebraic geometry to geometric properties of solid bodies in space and to special collections of polynomials. It is hoped that this work will lead to the development of some valuable new techniques in the field.