Resources

Below are some resources I compiled for students at UPenn interested in taking the major part of their oral exam in Complex Algebraic Geometry with Ron Donagi and Tony Pantev. 

1. The most important source: A list of problems that previous student compiled (link). 

2. Some oral exam transcripts in the official UPenn oral exam archive (link).

3. My exam related resources:

   • Practices problems (link) given to me by Tony Pantev before the oral. We practiced for about a year.

   • Practices problems (link) given to me by Ron Donagi before the oral. This was also our homework problems for my year long course in complex algebraic geometry (MATH 6220+6230) at Penn, taught by Ron Donagi.

   • My oral exam transcript (link), which includes some fun conversations before and after the exam, the actual problems on the exam, and some reflections I made after the exam was over. I wrote this after I finished my exam, so it was messy, sloppy, incomplete, probably does not make much sense, but I will not update it anymore because I am lazy.

4. Other resources I compiled:

   • My notes on complex algebraic geometry (link), based on my year long course in complex algebraic geometry (MATH 6220+6230) at Penn, taught by Ron Donagi.

   • My oral syllabus (link).

   • My review notes on curves (link), which is included in Chapter 7 of my complex algebraic geometry notes.

   • My oral exam review sheet (link).  

Learning the Material & Recommendation for Textbooks

If you want to do your major or minor of your oral exam (See the official website for more detail about the oral exam) in complex algebraic geometry with Ron Donagi and Tony Pantev, the material that will be tested can roughly be divided into two parts: (1) Complex manifolds/algebraic geometry, (2) Curves. In an oral exam conducted by Ron and Tony, each part usually constitutes roughly 50% of the problems. Your syllabus (which you have to make for yourself and it needs to be approved by Ron and Tony) should cover both of these parts. My syllabus attached in the previous section should be a more or less reasonable example.  Next I discuss how to study these materials and prepare for your oral exam (more on this later).

Coursework

At UPenn, we offer two year-long courses in algebraic geometry: the arithmetic version (more like Hartshorne style AG) and complex version (usually taught by either Ron or Tony) and they alternate every year. That means complex algebraic geometry (MATH 6220+ MATH 6230) is offered every other year. If you want to take the oral example on complex AG with Ron and Tony before the end of your second year, you should definitely take the complex algebraic geometry course (6220+6230) at UPenn before your oral exam.

Prerequisites:

The prerequisite for MATH 6220+6230 is mostly complex analysis, differentiable manifolds, and algebraic topology.  You can study these prerequisites by taking some (or all) of the first year graduate course sequence at UPenn. These corresponds to the first semester of the analysis course (MATH 6080, but only the complex analysis portion), the first year geometry and topology sequence (MATH 6000+6010). Make sure you take these courses or have equivalent background before you take 6220+6230, and make sure you understand these prerequisites REALLY WELL.

Schedule and Planning

Depending on whether 6220+6230 is available to you at your first year or second year at UPenn, your plan may differ slightly. The common thing is: it will only be avilable to you at either your first or second year, and when it is avilable to you, you have to take it.

1. If 6220+6230 is avilable at your first year: TAKE IT! The first year schedule should be 3 basic grad course sequence (analysis + algebra + geometry/topology), and choose 6220+6230 as your fourth course. You should already know some basic complex analysis from undergrad (if not, spend some time over the summer before your 1st year PhD and learn it). 

   • If you know all or most of the geometry+topology sequence (MATH 6000+6010), I encourage you to take the placeout exam for these courses and pass them (because these courses are a lot of work). In this case, you are extremeley lucky! Because this means you have sufficient background to understand relevant geometry and topology in 6220+6230, and since you are taking it in your first year, you have more than enough time to absorb the material well and prepare for your exam. 

   • If you don't know most of the geometry+topology sequence, things will be a bit harder because 6220+6230 assumes you know manifolds well enough. You should probably learn manifolds, tangent vectors, cotangent vectors, differential froms, and integration of differential forms by yourself quickly early on during the semester (if you can learn it over the summer before your first year it would be ideal). The first part of my notes on smooth manifold should be more than sufficient.

2. If 6220+6230 is available at your second year: TAKE IT! There are good and bad sides of taking it at your second year. The good side: you already know all the prerequisite pretty well by taking necessary first year grad course sequence at Penn. The bad side: It's a bit late to study the material. Your oral exam is only one year away. And you need to spend considerable amount of time doing problems with Ron and Tony (more on this in the next section). Therefore, my advice is:

Spend the summer of your first year learning complex algebraic geometry yourself and try to go through everything once!!!

You can use one of the textbooks that I recommend below. And in your second year, when you take 6220+6230 and it will really be a review and consolidation process for you, and you can focus on doing problems with Ron and Tony (More on this in the next section). 

Material and Textbook Recommendations

As mentioned, the material included in the oral exam can roughly be described by complex geometry and curves. My notes on complex algebraic geometry should cover most of them, and my syllabus is a decent reference. There are several math textbooks on this subject. I read quite a few and here are my recommendations (I know everyone has different tastes, so I am just sharing my biased opinion):

On complex (algebraic) geometry, there are 3 main references:

• Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry (Chapter 0-1, and the Chern class section in Chapter 3). This is a classic text. This is where all the old generation of complex algebraic geometers learned these material from. But to me, it is a bit disorganized, too dense, too many pages, and it is like a dense waterfall of continuous and heavy flow of knowledge. I tried reading a little bit of it, but it just didn't work for me. Also, be warned that it contains many typos and minor mathematical mistakes (in fact, one professor I knew became an expert in complex geometry just by finding all the typos and mistakes in Griffiths and Harris and correcting them!!!)

• Daniel Huybrechts, Complex Geometry: An Introduction. This is, overall, a fine book. I recommend getting this book and read it as a supplement along with some other books. I find many of the proofs in this book to be kind of dry, and Huybrechts present the material in a very weird order (like extracting some Lefschetz theory in chapter 1, put sheaf cohomology in the appendix, and have an overly condense discussion on Kahler manifolds). More over, who proves every isomorphism of vector bundles by proving that they have the same transition functions??? [Btw: his proof of the Euler sequence in section 2.4 is just garbage, do not look at that. See Griffiths and Harris p.408-409 for a way better geometric proof, or look at my notes :)]. But please do not get me wrong, this book has its value: The results that Huybrechts presents is the closest to what will actually be covered and tested in the oral exam. You should absolutely know of of the results this book has, but do not hesitate to look for better proofs if you feel necessary.

• Claire Voisin, Hodge Theory and Complex Algebraic Geometry I (Chapter 1-6). Some people like it. But I don't. It's such an intense treatment of the material in less than 150 pages. Voisin omits a lot of details in her proofs. AND THEY ARE NONTRIVIAL DETAILS!!! But anyway, if you like reviewing the material after the first round of study, it could be a good source to look into. Also, these two volume book that Voison wrote are probably one of the only texts out there that give a complete servey on modern Hodge theory. If you are interested in doing Hodge theory related stuff, do take a look at the later chapters and the second volume.

But allow me to give a (biased) recommendation for the GOATed textbook on this subject:

• John Lee, Introduction to Complex Manifolds. This book only came out in 2024, and I love every page of it. It is so detailed and clear, and so well written: Any human being can read it and understand complex geometry through it. However, it does not cover as much material (less than what is needed in the oral) as deeply. Therefore, If you are learning complex geometry for the first time, I strongly suggest starting with this book, and master all of it, then use Huybretchs as a second textbook.

On curves, here are some references I read (partially), and as usual I will give my recommendation at the end.

• Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry (Chapter 2, Oral exam covers: 2.1-2.3). Same complaint as before on this book. But this is a standard reference, and I'm pretty sure Ron and Tony learned curve theory from this book. In fact, they have the entire 800+ book engrained in their brains and they know it so well it becomes as natural as breathing to them. So, if you want a book that fit their way of thinking the most, this is definitely something you should check out.

• Rick Miranda, Algebraic Curves and Riemann Surfaces. I've heard many good things about this book. Unfortunately I haven't read it, but I briefly looked over the content and it covers all the curve material that Ron and Tony expect you to know very well. So I definitely encourage you to check it out.

• William Fulton, Algebraic Curves. I read the whole thing, but I didn't like it. Too dense, and not enough stuff covered. 

• Otto Forster, Lectures on Riemann Surfaces (Chapter 1-2). Very good book. Easy to read, nicely written, I would highly recommend. But still not enough stuff covered. Although these missing details can be very easily fixed by doing problems with Ron and Tony.

My recommendation:

• David Eisenbud and Joseph Harris, The Practice of Algebraic Curves (Chapter 1-6, 9). This is also published in 2024, a very new book. Funny enough, I found this book by accident in the math physics library when I was trying to borrow Miranda's textbook. I didn't know which one to read, so I read this one first and I absolutely love it [And I never read Miranda. Sorry Miranda :(] I found it unbelievable how well it aligns with what Ron and Tony expect me to know about curves that I didn't know back then. Very good book, a delight to read, many beautiful pictures. Go for it!

Doing Practice Problems (Your 2nd Year)

It is a tradition and an unspoken rule that if you want to do major with Ron and Tony on complex algebraic geometry, you need to start meeting with them regularly (~weekly or biweekly) roughly a year before your oral exam, and practice solving problems. Note that if you are taking MATH 6220+6230, you need to do this in addition to solving all the homework problem for the class. So this is a lot of work.

This process is THE MOST IMPORTANT component in your oral exam preparation, for several reasons:

1. I was told by some of their students that your performance during this year long process of doing problems completely determines whether you will pass the oral exam, and how you actually do in the oral exam does not matter if your overall performance during the practice session is good. I think this is a bit of an extreme statement, and one should take it with a grain of salt. But still, this should tell you how important the process is. And I do believe that if you practice for a year with them, you will not do too badly in the oral exam, and your chances of passing the exam is extremely high.

2. The problems that Ron and Tony ask have unique styles. They are very concrete, one can even say somewhat computational, but it is hard to solve unless you are already familiar with this type of problems, or have seen similar problems before. The good news is, they have a problem bank and they mostly ask the same type of questions. Over the year, graduate students of theirs documented the problems they got during practice sessions or actual exams (see resource section), and this should be your most important resource. If you have time, look through the problem bank they collected, look over the solutions. It helps a lot.

3. If you ask Ron and Tony to be in your oral comittee, chances are you want to work with one or both of them. This gives you a chance to know their style, decide whether you get along well, and most importantly: let them get to know you (if you ask a professor that does not know you very well to take you as their student, you have higher probability of getting rejected by them).

How does it work?

By the time you start your second year, you email both Ron and Tony (or talk to them), and ask to schedule meeting regularly to prepare for your oral exam. 

Personally, I ask them to give me problems to think about. I think about them for a week, solve some or all of them, type the solution in LaTeX. During the meeting, I present the solution to the problems that I solved, and ask about the problems I didn't solve (Usually there is no time for both in an hour long session, so you need to have priorities. I always prioritize asking what I don't know). Then near the end of the meeting, I ask them to give a new set of problems to solve (plus the ones I didn't solve previously, if any), and repeat the process.

I should also point out that it is completely fine to also ask them anything about complex algebraic geometry that you do not understand when you are learning. In other words, you can treat part of the meeting as some sort of "office hour" just for you if you like.

If you do this procedure, then roughly a month before your oral exam, you should have a LaTeX file containing both the questions they asked, and your solution. Mine is about 40-50 pages long, and I think anywhere from 30-60 pages should be a sign that you are doing decent amount of work (If you have below 20 pages, then maybe your proof is very concise, which is fine but just make sure that you do know all the details and can clearly reproduce all the required details in a problem on demand).

About 2-3 weeks before your oral exam, you will have your last few meetings with them. During the meeting, ask them to give you problems like you are in an actual oral exam, and solve the problems live in front of them. In other words, take these last few opportunities and enjoy several free mock oral exam!

After that comes the execution day.

Advices for Oral Exam

"Courage is grace under pressure". --Ernest Hemingway

I am a really bad exam taker, and I don't react well under stress. Here are some of my advices. It is very likely that you are a much better exam taker than I was, so you probably don't need my advice. Some of these are copied from my oral exam transcript.

• During your last meeting with Ron and Tony, politely ask whether they are willing to give you (some) problems that they will ask you during the oral exam, or if you have taken MATH 6220+6230 taught by them, ask them whether they are willing to ask (a subset of) homework problems in 6220+6230 during your oral exam. Because they might agree! I know they did it to some people (including myself). But I also know other people that didn't get questions from them beforehand. If they declined, that's completely fine: You can expect them to give you relatively manageable problems during the oral. If they agree to give you some problems beforehand, you are very lucky because that means you have a week to prepare what they will 100% ask you in the exam, and do not waste that opportunity!

• If Ron and Tony gave you problems beforehand and if they make it very clear that they will ask this during the exam, make sure your solutions to those problems are bulletproof. Having a week to prepare for problems that will actually appear on the exam and do them incorrectly is unacceptable. Practice telling these problems to a friend (several times if necessary) before the exam, and encourage them to interrupt you to ask anything they think are not clear, so that you can make sure you have the ability to explain your work under arbitrary level of scrutiny. Also, see if there are any obvious follow up questions (because examiners will very likely ask follow ups). If you don't know any, put the problems into AI like ChatGPT or Gemini and ask them to generate some follow up questions, and prepare them well.

• Boardwork: Divide the board into quarters, always write from left to right (the first quarter to the last quarter), and do not erase anything until you finish the last 1/4 of the board and erase the earliest board. It's good habit!!! And not just because nice board work is important, but also because in this way you do not lose any of your immediate prior work and you can quickly remind yourself by looking at it (Read my oral transcript and see how I learned this the hard way). Finally, I should not have to tell you this, but write cleanly and neatly, so that they can read. My oral exam boardwork (it is by no means excellent, and I made mistakes as I just said, but it's an example):

• Also, I find the oral exam to be extremely stressful personally. I could not think calmly and I could only autopilot and do some computations, or the problems I have seen before. So make sure to practice beforehand (ask a friend to simulate an oral, or practice live oral with Ron and Tony beforehand), get used to the stress, because it is not the same as you doing problems over the board by yourself and maybe even talking out loud (Unless you became a first year at Penn in fall 2025, then you have already had SO MANY oral exams for your first year classes, and you are probably a much better candidate than I was).

• Finally, be really fluent with the basics. And by basics I mean definitions, basic results (without proofs), and computations. By computations I mean: you should be comfortable computing common Picard groups, Hodge diamonds, do blow up calculations in local charts, any coordinate computations, Chern class calculations, Riemann-Roch, Riemann-Hurwitz, adjunction, etc. It is possible that they ask a problem in oral exam that requires a nontrivial idea to solve (unlikely), and you cannot get it because you are nervous and cannot think well. That's fine: the examiners will give you enough help. But your computation skill should be robust under pressure: Even if you are under immense stress and you cannot think clearly and logically, you should still be able to perform computations automatically. Your ability to do computations should not collapse under pressure.

Good Luck on Your Oral Exams!!!