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

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.