Riemann Roch Theorem: A Topological Perspective

[mathwonk]

we had a nice undergraduate version of the course on curves successfully taught recently using the book, plane algebraic curves, by gerd fischer.

an older out of print book, that is excellent, is fulton's algebraic curves. in particular, fulton proves all the main results on curves, bezout, resolution of singularities, and riemann roch.

 

[mathwonk]

Thats an interesting list of nice books, bombadillo.

If I was unclear, I was not recommending a book to learn curves from, merely indicating that hartshorne is not at all the level thought appropriate for undergrads by at least one famous algebraic geometer in britain.

I enjoyed perusing your list as well, a few of which i own. they are very diverse in nature, and certainly include some very elementary entry level books. I add a few remarks.



1) Conics and Cubics. Bix. Springer.

apparently extremely elementary, discussing only the curves in the title, although cubics are already a sophisticated topic, at the upper end of their study, having played the lead role in wiles solution of fermats last theorem.

2) Elementary Geometry of Algebraic Curves. Gibson. Cambridge.

another quite elementary treatment, again maxing out pretty much at the addition law on cubics it seems.

3) Plane Algebraic Curves. Brieskorn and Knorrer. Birkhauser.

an impressive tome of over 700 pages, with sections on history, elementary algebraic methods (resultants), and more sophisticated topological methods, resolution of singularities and integrals of differential forms, by a master researcher (Brieskorn) on the theory and practice of analyzing singularities. far froma self contaiend boiok, but very illuminating and interesting, he discusses some prettya dvanced topics such as hodge theory and algebraic derham hypercohomology, but seems never to even prove the riemann roch theorem. his approach is often to tell you the many facts he knows about the subject, and then just pull in whatever heavy machinery he needs to deduce them.

tha advantage of his un - self - contained approach is he tells you a lot you would not hear if he were restricted to what he can fully present with all the background.

4) An Invitation to Algebraic Geometry. Smith, et al. Springer.

apparently kind of an informal overview of what goes on in algebraic geometry, not restricted to curves really, nor an introduction to them.


5) Complex Algebraic Curves. Kirwan. Cambridge.

i could not get hold of a copy, but the author is a well known algebraic geometer, a student of Atiyah I believe. this book should be a good treatment.


6) Introduction to Commutative Algebra and Algebraic Geometry. Kunz. Birkhauser.
again not an introduction to curves, but a general introduction to commutative algebra and algebraic geometry.


7) Algebraic Curves and Riemann Surfaces. Miranda. AMS

this really is a thorough going introduction to curves over the complex numbers, with complete proofs of the main results, such as riemann roch theorem (a la Weil and Serre)

8) An Invitation to Arithmetic Geometry. Lorenzini. AMS.

a lovely introduction to the arithmetic side of the subject, the algebraic concept of integral closure, due to zariski, discriminants, ideal class group, and again reproducing the proof of Weil using ideles, (did Serre add anything?) for riemann roch.

so the book to choose depends on what you want out: you can have fun playing around with elementary conics and cubics, or go for a development that includes riemann roch, and hopefully also some significant applications, to jacobian varieties and projective models of curves, such as found in p. griffiths' china lectures, introduction to algebraic curves. he proves most of the basic results (bezout, riemann roch) and applies them too.

 

[bombadillo]

we had a nice undergraduate version of the course on curves successfully taught recently using the book, plane algebraic curves, by gerd fischer.

an older out of print book, that is excellent, is fulton's algebraic curves. in particular, fulton proves all the main results on curves, bezout, resolution of singularities, and riemann roch.

I have the Fischer book, and I have the Fulton book in storage in England. Another title which I may buy sight unseen is Hulek's Elementary Algebraic Geometry, published by the AMS. I'm also looking forward to Kunz's book on plane algebraic curves, which will be puiblished by Birkhauser, and due in a month or two.

I did attend a grad course on Riemann surfaces in London, but the nincompoop of a lecturer didn't even get to the Riemann-Roch theorem.

The problem with the British university system was its rapid growth during the 60s. At that time anyone with a PhD and one or two indifferent papers could get a lecturership at the rapidly expanding universities. Needless to say, most of these lecturers were no good. Once hired, it was next to impossible to chuck them out, and they've continued to be a blight on the English academic scene for decades. Brilliant young scholars have been unable to obtain academic appointments simply because these was no room anywhere. With these mediocre time servers as lecturers, the quality of undergraduate math education has continued to slide downhill, with the possible exception of Oxbridge and Warwick. Of course, this isn't the whole story: school-leavers are arriving at universities with less technical virtuosity than of old.

 

[mathwonk]

i am not an expert on riemann roch, but do appreciate it and am a huge fan of it. I have writtena set of notes on it with an account of riemanns own proof as wella s a sketch of serre weil's proof,a nd also a sheaf theoretic version in higher dimensions and an inmtroduction to hirzebruch's version. if you like i'll send you a pdf version if you send me your email.

it has a couple typos but not too many i hope.

to make a long story short, riemann roch is an analog of the mittag leffler theorem for compact non planar surfaces.

i.e. on a compact two dimensional surface with complex structure, by the residue theorem, given a non constant meromorphic function f, then for every holomorphic differential one - form w, the residue of the meromorphic differential fw is zero. conversely a set {Pi} of local principal parts forms the polar part of a global meromorphic function f if and only if for every holomorphic differential w, the residue of the system {Piw} is zero.


This converse of the residue theorem is almost the riemann roch theorem. that theorem goes further and counts the number of meromorphic functions with given polar behavior.

in higher dimensions the theorem is weaker, and only computes by topological data, the alternating sum of a sequence of dimensions of vector spaces, one of which is the number of meromorphic functions with given polar behavior. there is however a criterion for this alternating sum to equal the desired number, due to kodaira, which holds for high degree polar behavior, but not always.

the theorem has many consequences, e.g. every complex curve homeomorphic to a projective line is also complex holomorphically isomorphic to one, given any elliptic curve (curve of genus one) and any three points on it, there is an embedding of that curve in the complex projective plane as a cubic with those three points collinear. given any curve of genus 2 there is a realization of that curve as a double cover of the projective line with 6 branch points. any curve of genus three is either a smooth plane quartic curve or a double cover of the line with 8 branch points. etc...