Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This course is the second in a two-quarter introductory sequence that develops the basic theory of this classical mathematical field. Whereas the fall-quarter course focused more on the subject’s algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include Bézout’s Theorem, rational varieties, cubic curves and surfaces (including the remarkable 27-line theorem), and the connection between varieties and manifolds. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.

Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This course is the second in a two-quarter introductory sequence that develops the basic theory of this classical mathematical field. Whereas the fall-quarter course focused more on the subject’s algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include Bézout’s Theorem, rational varieties, cubic curves and surfaces (including the remarkable 27-line theorem), and the connection between varieties and manifolds. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.

Alright math nerds, it’s that time again! Be sure to register for Mike Miller’s excellent follow-on course for Algebraic Geometry.

Don’t forget to use the coupon code EARLY to save 10% with an early registration–time is limited!

I originally made this compilation on May 31, 2016 to share with some friends and never got around to posting it. Now that I’m actually in the midst of a class on the topic, I thought I’d dust it off and finally publish it for those who are interested.

If you’re aware of things I’ve missed, or which have appeared since, please do let me know in the comments.

A List of video lectures for Algebraic Geometry

Harpreet Bedi (YouTube) 68 lectures (Note: His website also has some other good lectures on Galois Theory and Algebraic Topology)

The Institute for Advanced Study is deeply saddened by the passing of Vladimir Voevodsky, Professor in the School of Mathematics.
Voevodsky, a truly extraordinary and original mathematician, made many contributions to the field of mathematics, earning him numerous honors and awards, including the Fields Medal.
Celebrated for tackling the most difficult problems in abstract algebraic geometry, Voevodsky focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and algebraic topology. He made one of the most outstanding advances in algebraic geometry in the past few decades by developing new cohomology theories for algebraic varieties. Among the consequences of his work are the solutions of the Milnor and Bloch-Kato Conjectures.
More recently he became interested in type-theoretic formalizations of mathematics and automated proof verification. He was working on new foundations of mathematics based on homotopy-theoretic semantics of Martin-Löf type theories. His new "Univalence Axiom" has had a dramatic impact in both mathematics and computer science.

Sad to hear of Dr. Voevodsky’s passing just as I was starting into my studies of algebraic geometry…

For those who are still on the fence about taking Algebraic Geometry this quarter (or the follow on course next quarter), here’s a downloadable copy of the written notes with linked audio that will allow you to sample the class:

I’ve previously written some notes about how to best access and use these types of notes in the past. Of particular note, one must download the .pdf file and open in a recent version of Adobe Acrobat to take advantage of the linked/embedded audio file. (Trust me, it’s worth doing as it will be like you were there with the 20 of us who showed up last night!)

For those who prefer just the audio files separately, they can be listened to here, or downloaded.

This is a genuine introduction to algebraic geometry. The author makes no assumption that readers know more than can be expected of a good undergraduate. He introduces fundamental concepts in a way that enables students to move on to a more advanced book or course that relies more heavily on commutative algebra.
The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory.
The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary. It is also an excellent text for those working in neighboring fields (algebraic topology, algebra, Lie groups, etc.) who need to know the basics of algebraic geometry.