Problems in Algebraic Geometry

Lazarsfeld将致力于代数几何中的一些问题。第一系列问题涉及射影簇的大度嵌入的代数性质。在过去的25年里，定义曲线的方程得到了深入的研究，但直到最近，人们对更一般的变种的代数行为知之甚少。在这个方向上，Lazarsfeld将继续他与Ein关于随着嵌入线丛的正性增长而研究高维变种合子的渐近结构的工作。Lazarsfeld还将研究一些关于高余维代数圈的正性性质的问题。由他与Debarre、Ein和Voisin的工作发展而来，这里的想法是探索余维为1的循环的经典正性概念的高余维类似。代数几何是数学中最古老和最核心的领域之一，它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有联系，从数论到拓扑学、代数和复数分析。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，Lazarsfeld将研究方程的性质，去掉固定的几何轨迹。希望这项研究将为代数几何和交换代数领域开辟新的接触点。Lazarsfeld还将致力于一些猜想，涉及较小的几何轨迹可以位于较大的几何轨迹中的方式。

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.