Algebraic geometry by Richard E Borcherds

Sobre este curso

Algebraic geometry is where equations become shapes and algebra becomes a tool for seeing hidden structure. This free online course guides you from classical plane curves to the modern language of varieties and schemes, helping you develop the kind of mathematical maturity that supports advanced study in algebra, number theory, and geometry. Along the way, you train yourself to move comfortably between geometric intuition and the precise algebra that controls it: ideals, coordinate rings, maps, and the way they encode spaces.

You begin with concrete problems that motivate the subject: parametrizing curves, understanding intersections, and working in projective space where points at infinity clarify what affine pictures miss. As the viewpoint widens, you learn how the Zariski topology organizes algebraic sets, why Noetherian conditions matter, and how Hilbert’s Nullstellensatz connects geometry over algebraically closed fields with the algebra of polynomial rings. These ideas are not treated as isolated facts; they gradually become a framework you can actually use to reason about dimension, regular functions, and morphisms between varieties.

A key strength of the course is the steady transition from classical algebraic geometry to tools that power contemporary research. You explore quotients by group actions and invariants, projective constructions like Grassmannians and bundles, and geometric operations such as blowups, resolutions, and flops that explain how spaces change under controlled modifications. Singularities are treated as objects you can diagnose and study via tangent spaces and local algebra, building an instinct for what goes wrong and how geometry repairs it.

The second half introduces schemes and sheaves, providing the unifying language needed to make sense of localization, gluing, Spec and Proj, and morphisms at a level that supports further coursework. You will see how properties like separatedness and properness fit naturally into the theory, and how divisors, line bundles, and the Picard group capture subtle geometric information. Exercises throughout encourage active understanding, so that you finish not only able to follow definitions, but also able to compute, explain, and apply the core ideas with confidence.