Free online courseAlgebraic geometry

Course Description

Explore the world of algebraic geometry through a comprehensive free online course that offers an in-depth look into the fundamental concepts and advanced topics. This course covers a broad spectrum of subjects, starting with an introduction to algebraic geometry and delving into the intricacies of cubic curves, Bezout, Pappus, and Pascal's theorems. Discover the fascinating world of Kakeya sets, affine space, and the Zariski topology, while gaining insights into Noetherian spaces.

Venture into the nullstellensatz, both weak and strong, and explore the profound implications of the Lasker Noether theorem through detailed proofs. Learn about the quotients of varieties by groups, and understand Hilbert's finiteness theorem, along with practical examples of quotients. Grasp the concept of dimension and projective space, and unravel the mysteries of Desargues's theorem and products of varieties.

Analyze the Veronese surface, Grassmannians, projective space bundles, and toric varieties. Delve into categories, regular functions, and morphisms of varieties, while exploring affine algebraic sets and commutative rings. Master the topics of products of projective varieties, automorphisms of space, and the Ax Grothendieck theorem. Understand the role of rational maps and the complexities of elliptic functions and cubic curves.

Investigate rationality of cubic surfaces, techniques of blowing up a point, and the Atiyah flop. Study singular points, Zariski tangent space, Du Val singularities, and explore examples of resolutions and completions. Learn about resultants, proper maps, and get a comprehensive survey of curves, including Hurwitz curves.

Engage with the resolution of curve singularities and Newton's rotating ruler. Discover Hilbert polynomials and the degree of a projective variety while revisiting Bezout's theorem. Transition into schemes with lessons on introduction, etale spaces, exactness and sheaves, and the definition of a scheme.

Comprehend morphisms of affine schemes, gluing schemes, the concept of Proj S, and the functor of points. Grasp irreducible, reduced, integral, and connected schemes, and understand properties like quasicompact and Noetherian morphology. Explore morphisms of finite or quasifinite type, immersions, products, and group schemes.

Delve into valuation rings, proper morphisms, projective varieties, and quasicoherent sheaves. Study invertible sheaves, coherent sheaves, line bundles on projective space, and vector bundles. Examine divisors on a Riemann surface, comparing Weil and Cartier divisors, and study the canonical sheaf and other advanced concepts.