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，它研究多元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中有着重要的应用。主要问题上的PI将工作涉及最近推出的措施的复杂性代数簇。其思想是定量地理解给定的代数轨迹在多大程度上不能由不满足多项式关系的坐标描述。这提供了一种新的方式来区分几何不同种类的解决方案集。 PI将研究代数几何中的一些问题。第一系列问题涉及代数簇的非理性的措施。虽然研究哪些品种是理性的或接近理性的是经典的，但最近人们对测量和控制一个给定的非理性品种的“非理性程度”的问题感兴趣。PI将继续他的工作与艾因和其他几个不变量在这个方向上，包括最低程度的合理覆盖的射影空间，和最少的gonality覆盖家庭的曲线。在另一个方向上，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型映射应用到代数曲线上的Koszul圈的向量空间来定义的。

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.