Algebraic Geometry Part II

Welcome to the course webpage for Part II algebraic geometry, taught in the 2023 Lent term at Cambridge. 

Instructor: Dhruv Ranganathan (E 1.01). Meeting time: MWF 12-1 PM (MR4)

Overview: The course will be an introduction to the basic theory of affine and projective algebraic varieties, as well as the theory of algebraic curves. These objects have been, and remain, the fundamental objects of interest in algebraic geometry. The material brings together ideas from algebra, topology, and geometry. We will try to learn the basic language, keep an eye on the richest examples of interest, all with a view towards modern developments.

Resources: I will provide TeX lecture notes at this link. The file will be updated roughly weekly, and will contain slightly more information and may frequently contain references and details that were not provided in lecture. There will also be four example sheets which will become available at the usual link

Example sheets: 1 2 3 4

There is no textbook that covers exactly the material that we will cover, and the TeX notes form the main resource. Texts are always good for different presentations however. In the first part of the course, texts of Hulek (Elementary Algebraic Geometry) or Reid (Undergraduate Algebraic Geometry) are excellent, as is Fulton (Algebraic Curves). The material on algebraic curves forms roughly the last third of the course, Fulton's text contains everything necessary, but Kirwan (Complex Algebraic Curves) is very insightful. 


Where to go next?

If you would like to go a little bit further in exploring algebraic geometry, I recommend trying to first gather up a good cluster of examples. Joe Harris's book "Algebraic Geometry: A First Course" is a great start. In order to solidify the technical foundations, the first chapter of Hartshorne's "Algebraic Geometry" should now be very accessible. The next significant new topic to study is the theory of schemes, which is the content of most of the above book by Hartshorne. If you're more geometrically inclined, you canm look at Donu Arapura's "Algebraic Geometry over the Complex Numbers".