The goal of the course is to make the student familiar with some of the basic concepts and techniques of modern algebraic geometry.
In its modern form, algebraic geometry uses the language of schemes. A first goal of the course is to introduce this language. The most convenient level of generality for doing this features (locally) ringed spaces, and Yoneda’s lemma. The geometry comes in when we define (affine, projective, ….) schemes and study various useful properties of morphisms of schemes. The notion of scheme generalizes that of a variety. The course finishes with a discussion of some cohomology of sheaves, and a discussion of Serre duality on curves over a field, leading to a proof of (a special but already very powerful case of) the Riemann-Roch theorem.