What to study after Miranda's "Algebraic curves and Riemann surfaces"? I am supervising a small reading group on Riemann surfaces. We are following Rick Miranda's book "Algebraic curves and Riemann surfaces". We will probably be done at the end of the year, and we would like to continue the seminar. What would be the next best thing to study ?
The students are undergrad, so they know topology, algebra, complex analysis and multivariable calculus. We will also, roughly, be familiar with most of the book. (One of the students really wants to study sheaf theory, so something with some sheaf theory would be nice.) They don't know algebraic geometry (other than what is in Miranda).
I have some ideas of course, in particular "chapter on algebraic surfaces" by Miles Reid, and "Hodge theory and complex geometry I" by Claire Voisin. But that might be too hard just after Miranda, so I am interested in other propositions. If possible, avoid suggestions like reading Hartshorne (it's a lot of heavy machinery, and for example most of the applications of chapter 4 can be obtained by elementary methods over $\mathbb C$, like in Miranda's book.)
 A: One interesting (albeit potentially quite hard) direction might be to stick with curves and go deeper into their geometry. The standard text here is Geometry of Algebraic Curves, Volume 1 by Arbarello, Cornalba, Griffiths, and Harris, but frankly with undergrads you could probably spend quite a long time just on the first chapter on Preliminaries (this is not a bad thing; there is a lot of geometry in the chapter, plus a ton of exercises).
One good way to trim down the material in the book would be to follow the notes from around 2011 when Joe Harris taught a course on the subject. There are several sets of typed notes easily found on Google.
A: Miles Reid is a good idea. I also recommend Shafarevich's Basic Algebraic geometry - perhaps after already covering Miranda you might want to pick and choose the chapters you read.
A: Huybrechts' book titled Complex Geometry and Griffiths Harris 'Principles of algebraic geometry" chapters 0 and 1
Source:
What are some alternatives to Griffiths Harris 'Principles of algebraic geometry" chapters 0 and 1?
