Algebraic Geometry is the study of the geometry of solution sets of systems of polynomial equations. This is a central area of modern mathematics with deep connections to Representation Theory, Number Theory, and many other branches of mathematics, as well as applications to a broad spectrum of areas including cryptography, theoretical physics, and computer science.
The goal of the course is to introduce the basic notions and techniques of classical algebraic geometry using only a minimum of required prerequisites. It covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. We will show how ideas from linear algebra and ring theory can be applied to yield geometric insights.
Curves and surfaces, Affine and projective varieties, Zariski topology, Hilbert Nullstellensatz, Bézout’s theorem, Tangent space and smoothness, irreducibility and dimension, morphisms, Grassmannian and flag varieties, determinantal varieties, Normal varieties, Birational equivalence and Zariski’s main theorem, Resolution of singularity, Divisors and Riemann-Roch Theorem for curves.
A course in Ring theory is required. A review of relevant definitions/theorems from commutative algebra will be given either at the start or as needed during the course.