6/30/2023 0 Comments Algebraic geometry examples![]() ![]() However, I want to remake a point I made on my blog, which is that later on in your career you will be much better at learning things than you are now. I'm not an algebraic geometer, but I do know several algebraic geometers and it's clear that modern algebraic geometry is a very large field some aspects of which involve technical modern abstractions (stacks!) others of which are in a more combinatorial direction (toric varieties, Grobner bases) and others involve more classical algebraic geometry. ![]() ![]() ), they are problems of geometry, not algebra, and there are many available avenues to approach them: algebra, analysis, topology (as in Hirzebruch's book), combinatorics (which plays a big role in some investigations of Gromov-Witten theory, or flag varieties and the Schubert calculus, or. ![]() The central questions of algebraic geometry are much as they have always been (birational geometry, problems of moduli, deformation theory. In summary, if you enjoy commutative algebra, by all means learn it, and be confident that it supplies one road into algebraic geometry but if you are interested in algebraic geometry, it is by no means required that you be an expert in commutative algebra. To understand Siu's work you will also need to learn the analytic approach to algebraic geometry which is introduced in Griffiths and Harris. Rather, you will need to learn geometry! And by geometry, I don't mean the abstract foundations of sheaves and schemes (although these may play a role), I mean specific geometric constructions (blowing up, deformation theory, linear systems, harmonic representatives of cohomology classes - i.e. Indeed, they lie squarely on the same axis of research that the Italians, and Zariski, were interested in, namely, the detailed understanding of the birational geometry of varieties.įurthermore, to understand these results, I don't think that you will particularly need to learn the contents of Eisenbud's book (although by all means do learn them if you enjoy it) Both these results would be of just as much interest to the Italians, or to Zariski, as they are to us today. However, the questions being studied are (by and large) the same.Īs I commented in another post, two of the most important recent results in algebraic geometry are the deformation invariance of plurigenera for varieties of general type, proved by Siu, and the finite generation of the canoncial ring for varieties of general type, proved by Birkar, Cascini, Hacon, and McKernan, and independently by Siu. The only distinction to me seems to be chronological: modern work was done recently, while classical work was done some time ago. I agree with Donu Arapura's complaint about the artificial distinction between modern and classical algebraic geometry. Is this right? If not, succinctly my question is: how great an influence does classical algebraic geometry have on modern algebraic geometry today? While I suspect that, as with other branches of mathematics, "abstraction was invented to analyze the concrete", with all the emphasis currently given to the understanding of abstract tools, for someone who is not very familiar with the subject (such as myself), it seems that algebraic geometry is a "mixture" of general topology and abstract algebra. Would you need to be familiar with something like the contents of Eisenbud's Commutative Algebra: With a View Toward Algebraic Geometry, or is less needed in reality? (I am familiar with more commutative algebra than that which is covered in Atiyah and MacDonald's *Introduction to Commutative Algebra", but less than that which is covered in Eisenbud's textbook.)Īlso, is modern algebraic geometry concerned with abstractions such as schemes, sheaves, topological spaces, commutative and noncommutative rings etc., or is it just classical algebraic geometry in an abstract form? Perhaps more specifically, to do research in modern algebraic geometry, do you need to be familiar with classical algebraic geometry, or is it possible to think of algebraic geometry as an "abstract language" and do research based just on this perception? one needs to know to do research in (or to learn) modern algebraic geometry. Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. ![]()
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