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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.)

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  • $\begingroup$ Can you provide some more information about what "best" means here and what direction you hope this goes? As currently written, this seems rather open-ended to me. $\endgroup$
    – KReiser
    Commented Oct 10, 2020 at 9:05
  • $\begingroup$ @KReiser : thanks for your interest ! Indeed, the question is a bit open-ended. I was hoping for suggestions of reference of books 1) accessible after Miranda's book. 2) that are using (if possible) sheaf theory/line bundles. $\endgroup$
    – nicolas
    Commented Oct 10, 2020 at 9:27
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    $\begingroup$ Do you have feelings about where in the spectrum from complex geometry to schemes you'd like to be? Next, sheaf theory (in the sense of a basic introduction, the usual functors, etc) is rather pedestrian, but sheaf cohomology is pretty great - are you actually looking for just sheaf theory, or are you willing to tack towards cohomology? $\endgroup$
    – KReiser
    Commented Oct 10, 2020 at 9:37
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    $\begingroup$ @KReiser : The goal will certainly be to use sheaf cohomology ! (Actually, there is some sheaf cohomology in the last part of Miranda's book. I plan to prove Riemann-Roch and Abel-Jacobi by sheaf theory, so they will already know something). Do you have anything in mind ? For the spectrum : if possible something non-arithmetic (because I'd like to avoid heavy commutative algebra). I'd be interested by anything touching deformation theory, enumerative geometry, birational geometry, Hodge theory, moduli spaces, vector bundles on curves, ... $\endgroup$
    – nicolas
    Commented Oct 10, 2020 at 11:24
  • $\begingroup$ Huybrechts' book titled Complex Geometry Source: math.stackexchange.com/questions/3888577/… $\endgroup$
    – BCLC
    Commented Nov 4, 2020 at 6:09

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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.

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  • $\begingroup$ Thank you ! I just read the notes by Akhil Matthew on Harris class. This is a very good suggestion but actually Miranda's book covers several advanced topics in curves as well (introduction to Brill-Noether theory, Flex/Inflection/Weierstrass points, ...), so we'll probably pick something else. $\endgroup$
    – nicolas
    Commented Oct 12, 2020 at 7:41
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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.

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  • $\begingroup$ Thank you ! This is a better suggestion than Hartshorne but if we will choose to read a textbook in algebraic geometry, I'd rather go for Harris's book that has more example. $\endgroup$
    – nicolas
    Commented Oct 12, 2020 at 7:29
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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?

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