Research in Algebraic Geometry: Irrationalty of Algebraic Varieties and Koszul-Wahl Cohomolgy Groups

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。代数几何在编码理论、理论物理和计算数学等不同领域中都有重要的应用。PI将解决的主要问题涉及最近引入的代数簇复杂性的测量。其思想是定量地理解给定的代数轨迹在多大程度上不能用不满足多项式关系的坐标来描述。这为在几何上区分不同种类的解集提供了一种新的方法。PI将解决代数几何中的一些问题。第一系列问题涉及代数簇的无理性度量。虽然研究哪些品种是理性的或接近理性的是经典的，但最近人们对衡量和控制给定的非理性品种的“非理性程度”的问题感兴趣。PI将继续他与Ein和其他人在这个方向上的几个不变量的工作，包括射影空间的有理覆盖的最小次数，以及覆盖曲线族的最小正交性。在不同的方向，PI将探索一些新的上同调群的几何意义，这些上同调群是通过将Wahl型映射应用于代数曲线上的Koszul圈的向量空间而定义的。

Algebraic geometry, one of the oldest and most central fields of mathematics, studies the geometric properties of solutions to systems of polynomial equations in several variables. It has connections with many other fields of mathematics, ranging from number theory to topology, algebra, and complex analysis. Algebraic geometry has found important applications to problems in such diverse areas as coding theory, theoretical physics and the mathematics of computation. The main problems on which the PI will work involve recently introduced measures of the complexity of algebraic varieties. The idea is to understand quantitatively to what extent a given algebraic locus fails to be describable by coordinates that satisfy no polynomial relations. This provides a new way of distinguishing geometrically between different sorts of solution sets. The PI will work on a number of problems in algebraic geometry. The first series of questions concerns measures of irrationality for algebraic varieties. While it is classical to study which varieties are rational or nearly so, there has been recent interest in the problem of measuring and controlling "how irrational" a given non-rational variety is. The PI will continue his work with Ein and others on several invariants in this direction, including the least degree of a rational covering of projective space, and the least gonality of a covering family of curves. In a different direction, the PI will explore the geometric meaning of some new cohomology groups defined by applying Wahl-type maps to the vector spaces of Koszul cycles on an algebraic curve.