Let be a curve over a field and a divisor on . The Riemann-Roch theorem is a very precise statement about the dimension of , roughly saying that . There is a weaker version which I like to call “weak Riemann-Roch”, which is good enough for many purposes and much more elementary. In this post, I will state and prove that version.

So, definitions. Let be a field. Let be an integral regular projective curve over . Since we have said “regular”, “integral” is the same as “irreducible” is the same as “connected”. Since we have said “curve”, “projective” is the same as “proper”. If the field is perfect, “regular” is the same as “smooth over “.

Recall that, for a point of , we write for the residue field at . For a divisor , we put . A basic lemma is that, if and are rationally equivalent, then .

Theorem (Weak Riemann-Roch): With the above definitions, there is a constant such that, for all divisors , we have .

Let be a qcqs scheme and let be an invertible sheaf on . In class, we defined to be ample if the set of non-vanishing loci of sections of is a basis for the topology of . I tried to give a quick proof that this condition implies that is separated, but my proof had some gaps. I’ll fix them now, and also repeat the part that was right.

The “Algebraic Hartog’s Lemma” is Lemma 13.5.11 in Ravi’s book. Slightly restated, it says the following:

Let be an integrally closed Noetherian integral domain and let . Let . The following are equivalent:

(1) is in .

(2) For all dvrs with , we have .

(3) For all codimension one primes , we have .

The implications are straightforward, but I find quite confusing. And I have to say that Ravi’s proof doesn’t help me; it feels like a lot of pushing around modules that I don’t follow. So I put in a fair bit of time trying to find my own route and wound up frustrated: I thought I was making a fair bit of progress, but then I get stuck at the end.

I’m very glad to hear about other proofs that people like!

The following confusing issue came up in class. Theorem 15.2.2 says:

For a fixed scheme , maps are in bijection with the data , where is an invertible sheaf and are sections of with no common zeroes …, up to isomorphism of this data.

There is then a parenthetical remark: “(This works over , or indeed over any base.)”.

So, here was the confusing issue. Let be a base and let be an -scheme. We can talk about maps in the category of -schemes, and this is an additional condition: There are maps as schemes which are not maps in the category of -schemes. So on the “maps to projective space” side of the correspondence, matters.

However, the data of don’t appear to mention at all! In class, we considered whether there might be a notion of “invertible sheaf in the category of -schemes” or “section in the category of -schemes”, and we eventually decided that there was no room for in either of these definitions. So we were left puzzled.

I believe that I have now sorted it out. (Please tell me if still I’m wrong!)

In class we have been talking about line bundles and vector bundles, and a student asked the question “so, are the fibers of a vector bundle vector spaces?” I wanted to say “yes”. In a context where we weren’t focusing on foundations, I would just say “yes”. The purpose of this post is to explain that the answer morally is “yes”, but that there is a reason to hesitate.

In earlier editions of Ravi’s book (up until November 2017), quasi-coherent sheaves didn’t appear until Part V. This feels natural to me as a teacher: Parts II-IV (which were covered in the Fall term) are mostly about schemes, and Part V is mostly about sheaves, so this belongs in Part V. I assume that there was a reason that this material was moved earlier, to the end of Part II, but pedagogically, it makes sense to me that Aaron Pixton decided to skip it and that I am covering it now.

I required you all to go back and do the exercises in Section 4.1. There is a very nice remark here, that the proof of Theorem 4.1.2 is a “partition of unity” argument. I’d like to draw that out for you. I also had some requests to talk more about surjectivity. So here that is.

Algebraic Geometry is a field which has rewritten its foundations several times, and is still doing so now. Ravi’s book starts at a point deep in that multi-century long conversation, and moves the reader faster by disregarding what has come before. This is a choice which works, but I would like to spend at least a little time giving a bigger picture of the various answers there can be to the question “what is an algebraic variety” and, thus, to “what is algebraic geometry”?

Hello everyone! This is a blog associated with the course Math 632, Algebraic Geometry II, at the University of Michigan. The main course webpage is here. I’m the professor, David Speyer.

I’ve taught Algebraic Geometry before, but this is my first time teaching from Ravi Vakil’s textbook, Foundations of Algebraic Geometry. Ravi’s textbook was written in public: Way back in 2010, Ravi posted a note saying that he was working on an algebraic geometry textbook and seeking feedback on his ongoing drafts. For the last thirteen years, the mathematical world has gotten to watch Ravi’s book grow and has gotten to watch both students and experts discuss the best way to teach the field in the comments of Ravi’s blog.

Usually, when I teach advanced graduate courses, I assign my students to take notes on my lectures and I compile them into a single file. Here are what my students wrote the last time I taught Algebraic Geometry (first term, second term). This term, I am not going to do that, because I think it would come too close to simply telling students to rewrite Ravi’s book.

I also, this term, am going to expect students to take the reading assignments more seriously than ever before. In the past, I have assigned reading from Shavarevich and Hartshorne’s books, but I didn’t have the confidence to allow some topics to appear only in the readings without giving my own presentation of them in lecture. If I am going to keep up the pace that Aaron Pixton set in last term’s course, I am going to need to trust that there are some topics that Ravi teaches well enough that I will not need to cover them in class. To this end, I am instituting two innovations.

First, every week, I will have a google poll, due Friday, for students to ask questions, make comments or raise concerns about the reading. I promise that I will read and concern all the poll results before I come in for my first class of the week on Tuesday. Note that this poll is limited to enrolled students.

Second, I plan to write a blogpost here every week, giving my thoughts on the reading. My primary audience is the students enrolled in this class. However, I admit that I also have a secondary audience in mind; I hope that Ravi will, at some point, take a look through my thoughts. Note that the comments are open here on the blog: Please share your thoughts as well, and I suspect Ravi will also be interested in them!

Some students might wonder why this blog isn’t on Canvas. Canvas creates walled gardens, where our courses are hidden away from the world and are hard to find after the term ends. That can be good, for creating a safe space for discussion, or for protecting student privacy. But it also means that lots of great teaching is hidden away where the world can’t see it. I don’t know if I will do a great job, but I want to put my work here in public, where everyone interested in Algebraic Geometry can learn from it. To that end, if you are interested in Algebraic Geometry — anywhere in the world, and even long after this course has ended — I encourage you to read this blog, and I’ll be glad to have you engage in the comments, respectfully and in an on topic manner. Let’s see if we can make this a little fraction of the treasure that Ravi’s blog has been!