Algebraic Geometry

Algebraic geometry is the study of algebraically defined geometric objects, such as algebraic varieties (roughly, zero sets of systems of polynomial equations in many variables, over an algebraically closed field) or schemes (spaces which locally “look like” the prime spectrum of a commutative ring. It has a close relationship with commutative algebra, differential geometry, complex analysis, number theory, and representation theory.

The goal of the course is to introduce the audience to the basic objects of study in algebraic geometry: algebraic varieties and schemes, and (sheaves of) modules over them. Emphasis will be put on concrete objects such as projective varieties. In order to keep the course focused, we will work toward a single main goal, which will be the proof of the Weil conjecture (Riemann hypothesis) for curves over a finite field.