Geometry of complex projective varieties

This is a preliminary lecture note for the algebraic geometry course at the Second International Undergraduate Mathematics Summer School held at Tokyo University, July 29-August 9, 2019. It is of course impossible to teach algebraic geometry in five 90-minute lectures. Thus this would not be what I will try to do. Indeed, the goal of these lecture notes is to convey to the students a sense of what kind of objects/problems we study in algebraic geometry. Modern algebraic geometry requires the full machinery of Grothendieck: the language of schemes, sheaf and cohomology, and so on. On the other hand, algebraic geometry deals with polynomials, which are quite explicit. Quite often, explicit computations illustrate the general machinery. So I choose to (and probably have to) follow the low-tech approach and do things in an explicit way, giving hints to a general theory, if possible. In these lectures, I start with some classical examples of projective varieties, and then whenever I introduce a new concept, I study this concept with these examples in some details. Somewhat surprisingly, one can study these examples and prove interesting theorems rigorously with a minimum amount of prerequisite (one the level of what a good undergraduate student in his/her second or third year has learned), and some hard work. It is my hope that through the course, the students could get a rough idea of some of the basic ways of thinking about a problem in algebraic geometry. I assume the students are familiar with differentiable manifolds, vector bundles, some basic topology and abstract algebra. But to really understand the lectures, certain level of the mysterious “mathematical maturity” is absolutely essential. To avoid technicalities (more precisely, to make the students feel that they are in their comfort zone of differentiable manifolds), I will work over the complex numbers, although for the most part, I did not use anything special about complex manifold. I will try to make this course as self-contained as possible. But with Google and Wikipedia (or similar websites), I think the students should be able to catch up if there are some minor points that they have not encountered previously. Some helpful references include: [Har95, Mum76, Rei88]. The book by Harris contains lots of invaluable classical examples and exercises. The book of Reid has the same flavor as this note ( I found out this book while preparing the lectures and couldn’t agree more with the general philosophy in that book) and Chapter 8 is good to read for students interested in going further. Finally it is always enjoyable and enlightening to read a textbook by a master in the field, especially when the author is a good writer like Mumford.