Problems in Algebraic Geometry

拉扎斯菲尔德将研究代数几何中的一些问题。第一个系列的问题是关于投影变量的大次嵌入的代数性质。在过去的25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，人们对更一般的曲线变量的代数行为知之甚少。在这个方向上，拉扎斯菲尔德将继续与Ein一起研究随着嵌入线束正性的增长，高维变量协同的渐近结构。Lazarsfeld还将研究一些关于高维代数循环的正性的问题。从他与Debarre， Ein和Voisin的合作中发展出来，他的想法是探索余维数为1的循环正性的经典概念的高余维类比。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。它与数学的许多其他领域有联系，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的问题上也有重要的应用。在第一个系列问题中，Lazarsfeld将研究切割固定几何轨迹的方程的性质。希望本研究能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小几何轨迹可以位于大几何轨迹内的方式的猜想。

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.