Modern algebraic geometry vs. classical algebraic geometry - MathOverflow

Modern algebraic geometry vs. classical algebraic geometry

2010-07-13T02:04:00Z

Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. one needs to know to do research in (or to learn) modern algebraic geometry. 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.)

Also, 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?

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. Is this right? If not, succinctly my question is: how great an influence does classical algebraic geometry have on modern algebraic geometry today?

Answer by Daniel Litt for Modern algebraic geometry vs. classical algebraic geometry

2010-07-13T03:43:22Z

Should one learn point-set topology before real analysis or before studying metric spaces a bit? There are some advantages to doing so -- a more unified approach to real analysis or the study of metric spaces, for example. But this comes at the cost of all motivation for point set topology.

One can do "classical" algebraic geometry rigorously, and this is not a bad idea. It provides much-needed motivation for the language of schemes, sheaves, etc., which can otherwise seem incredibly unmotivated. And it generates intuition (which, to be fair, is often wrong) about these complicated and often pathological objects. But an even better reason to study classical algebraic geometry is to discover why Grothendieck, Serre, etc. wanted to come up with modern algebraic geometry in the first place; it's because classical algebraic geometry is so obviously in need of fixing. It's a beautiful subject, but I think it's pretty obvious from even a short study of it that you're not getting the whole story. (Bezout's theorem is a great example.) A good book to read if you want to get this feeling is Harris's "Algebraic Geometry, A First Course" -- it's a well-written book filled with great motivation, but you can't help but think that it's holding something back.

As for commutative algebra -- I think it makes sense to learn it concurrently with the geometry, which motivates it in an incredibly compelling way. Eisenbud is a good place to go for that kind of motivation.

I'm sure others will address the issue of doing research in the subject; I'm not qualified to comment on that, beyond mentioning that there's a wide variety of subjects researched in the field. Many people still work on subjects that might be considered "classical."

Answer by Noah Snyder for Modern algebraic geometry vs. classical algebraic geometry

2010-07-13T04:39:08Z

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.

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. As a result it's counterproductive to worry too much about what you should be learning now to maximize your efficiency of learning. Instead you should prioritize things you can learn now and which you enjoy learning now. Certainly you should start with an introductory algebraic geometry book, but once you're done with that there's no harm in looking at Eisenbud and seeing if you enjoy it. But if it's too hard going or if you feel like you're not fully appreciating it then go ahead and try reading something totally different. There'll be plenty of time to learn more commutative algebra while you're a grad student!

Answer by Francesco Polizzi for Modern algebraic geometry vs. classical algebraic geometry

2010-07-13T08:09:31Z

One of the main themes of "modern" algebraic geometry is the study of $families$ of algebraic varieties; in fact, just consider the huge subjects known as deformation theory and moduli spaces theory. This leads very naturally (at least from our "modern" point of view) to situations where the knowledge of schemes (non-reduced structures), commutative algebra (flatness, Cohen-Macauleyness, etc) and homological algebra are essential not only from a theoretical point of view, but also in order to make explicit computations with very concrete objects, e.g. quasi-projective varieties. Such "modern" tools have undoubtedly made the study of algebraic geometry harder for the beginner, but on the other hand they have brought clarity in many situations where the "classical methods" did not work well.

Answer by Changwei Zhou for Modern algebraic geometry vs. classical algebraic geometry

2010-07-13T10:36:10Z

I think one should work on modern ones. You can always pick up classical ones if you become suddenly interested in the realization of the abstract ideas. For books of course you need to read Hartshrone. I would say Eisenbud is not appropriate if you don't have other books to read. Anyway enjoy the books you read, and think hard before throwing away the books.

Answer by Roy Maclean for Modern algebraic geometry vs. classical algebraic geometry

2010-07-13T11:52:30Z

Depends what you mean by "modern".

"Numerical algebraic geometry" using homotopy methods is modern but concrete.

"Introduction to numerical algebraic geometry", A Sommese, J Verschelde, C Wampler http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.84.8283&rep=rep1&type=pdf

"The numerical solution of systems of polynomials arising in engineering and science", Andrew John Sommese, Charles W. Wampler, World Scientific, 2005

Answer by Donu Arapura for Modern algebraic geometry vs. classical algebraic geometry

2010-07-13T15:25:13Z

I think that some of the answers so far are very good, so this is a bit redundant. I just wanted to emphasize that the distinction between "classical versus modern" algebraic geometry is, to me, not a good one. While it's true that in scheme theory one encounters new phenomena, it can also be used to repair and extend the classical picture. For me, at least, the most beautiful parts of Hartshorne's book are the chapters on curve and surface theory.

As for background, I think that if you read Atiyah-Macdonald and do the exercises, you should be in pretty good shape to get started. I also usually tell my students to learn something about basic manifold theory and algebraic topology, since it provides some useful intuition for a number geometric/homological constructions.

Answer by Emerton for Modern algebraic geometry vs. classical algebraic geometry

2010-07-13T16:22:14Z

I agree with Donu Arapura's complaint about the artificial distinction between modern and classical algebraic geometry. The only distinction to me seems to be chronological: modern work was done recently, while classical work was done some time ago. However, the questions being studied are (by and large) the same.

As 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. Both these results would be of just as much interest to the Italians, or to Zariski, as they are to us today. 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.

Furthermore, 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); 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. Hodge theory, ... ). To understand Siu's work you will also need to learn the analytic approach to algebraic geometry which is introduced in Griffiths and Harris.

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.
The central questions of algebraic geometry are much as they have always been (birational geometry, problems of moduli, deformation theory, ...), 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 ... ), and who knows what others.

Answer by roy smith for Modern algebraic geometry vs. classical algebraic geometry

2011-05-12T20:34:16Z

For several beautiful and expert discussions of the contrasts and relations between classical and the evolving subject of abstract or modern algebraic geometry, I recommend the following ICM lectures:

O.Zariski, 1950, vol.2, p.77ff; B.Segre, 1954, vol.3, p.497ff; J.P.Serre, 1954, vol.3, p.515ff; A.Weil, 1954, vol.3, p.550ff; A.Grothendieck, 1958, p.103.

(This falls obviously under the heading "reading the masters".)

Indeed the whole algebraic geometry session, 1954, vol.3, pp.445-560, has an incredible list of short talks, (Groebner, Hirzebruch, Kodaira, Neron, Rosenlicht, Van der Waerden,...).

the link is: http://www.mathunion.org/ICM/

My apologies for such a brief answer.
The article by Zariski, THE FUNDAMENTAL IDEAS OF ABSTRACT ALGEBRAIC GEOMETRY, points out the advances in commutative algbra motivated by the need to substantiate results in geometry. “The past 25 years have witnessed a remarkable change in the field of algebraic geometry, a change due to the impact of the ideas and methods of modern algebra. What has happened is that this old and venerable sector of pure geometry underwent (and is still undergoing) a process of arithmetization. This new trend has caused consternation in some quarters. It was criticized either as a desertion of geometry or as a subordination of dis- covery to rigor. I submit that this criticism is unjustified and arises from some misunderstanding of the object of modern algebraic geometry. This object is not to banish geometry or geometric intuition, but to equip the geometer with the sharpest possible tools and effective controls.”

That by Segre argues for the preservation of geometric intuition in algebraic geometry for just this reason, for motivating and suggesting new questions to investigate. It seems particularly articulate and impassioned as he is arguing for a tradition that seems threatened to be lost.

GEOMETRY UPON AN ALGEBRAIC VARIETY BENIAMINO SEGRE I. Algebraic geometry — that is to say, the branch of geometry which deals with the properties of entities represented by algebraic equations — has in recent years developed in two distinct directions, which in a sense are opposed to one another. One of these directions is called abstract in as much as it is concerned with algebraic equations defined over commutative fields subject only to slight restrictions; here the means employed are purely algebraic, including in particular ideal theory and valuation theory. The other direction may properly be called geometrical) this usually deals with algebraic equations in the complex domain, and from time to time appeals to ideas and methods of analytic and projective geometry, topology, the theories of analytic functions and of differential forms. The dualism between these two disciplines has close relationship and affi- nity with that which, three centuries ago, arose between l'esprit géométrique of Descartes and l'esprit de finesse of Pascal, and which, in the past century, on the one hand divided the geometers into analysts of the school of Plücker and synthesists of the school of Steiner and, on the other, the algebraists into purists à la Dedekind and arithmetizers à la Kronecker. However, this dualism, instead of proving harmful to geometry, offers undoubted advantages when the two lines of development, with their respective merits and possibilities, are regarded not as contrasting but as complementary. We cannot fail to recognise in the abstract method and its technique a peculiar elegance, an impeccable logical coherence, and to appreciate the im- portance of the results so far obtained by it, particularly in the study of the foundations of geometry and the difficult questions concerning the singularities of algebraic varieties. But equally we cannot fail to recognise that the geometr- ical approach, with its greater concreteness, lends itself better to the formula- tion and initial study of new concepts and problems; and that it presents an incomparable wealth and colour of its own, due to the interweaving of many diverse strands, to the subtle and perspicuous play of geometrical intuition, and to the possibility of readily constructing examples and investigating special cases. We may also point out that, in the geometrical discipline, corresponding to a more definite notion of algebraic variety, there is a much wider range of subjects and a far greater number of orientations and contacts with other important branches of mathematics, which have found, and are finding, therein inspiration and extensions beyond the purely algebraic field.

Weil’s article describes how arithmetic benefits as well from the algebraization of geometry.

ABSTRACT VERSUS CLASSICAL ALGEBRAIC GEOMETRY ANDRé WEIL The word "classical", in mathematics as well as in music, literature or most other branches of human endeavor, may be taken in a chronological sense; it then means anything which antedates whatever one chooses to consider as "modern", and may be used to describe remote antiquity or the achievements of yesteryear, according to the mood and the age of the speaker. Sometimes, too, it is purely laudatory and is applied to any piece of work which is thought to be of permanent value. Here, however, while discussing algebraic geometry, I wish to use the words "classical" and "abstract" in a strictly technical sense which will be explained presently. Until not long ago algebraic geometers did their work exclusively with reference to the field of complex numbers; at the same time they worked on non-singular models, or at any rate their concern with multiple points was merely in order to try to push them out of the way by suitable birational trans- formations. Thus transcendental and topological tools of various kinds were available, and it was merely a matter of individual taste, personal inclination or expediency whether to use them or not on any given occasion. The most deci- sive progress ever made in the theory of algebraic curves was achieved by Riemann precisely by introducing such methods. Later authors took consider- able pains to obtain the same results by other means. In so doing, they were motivated, at least in part, by the fact that Riemann had given no justification for Dirichlet's principle and that it took many years to find one. Similarly, the use of topological methods by Poincaré and Picard, not to mention some more recent writers, has often been such as to justify doubts about the validity of their proofs, while conversely it has happened that theorems which had merely been made plausible by so-called geometrical reasoning were first put beyond doubt by the transcendental theory. Now we have progressed beyond that stage. Rigor has ceased to be thought of as a cumbersome style of formal dress that one has to wear on state occasions and discards with a sigh of relief as soon as one comes home. We do not ask any more whether a theorem has been rigorously proved but whether it has been proved. At the same time we have acquired the techniques whereby our prede- cessors' ideas and our own can be expanded into proofs as soon as they have reached the necessary degree of maturity; no matter whether such ideas are based on topology or analysis, on algebra or geometry, there is little excuse left for presenting them in incomplete or unfinished form. What, then, is the true scope of the various methods which we have learnt to handle in algebraic geometry? The answer is obvious enough. Let us call "classical" those methods which, by their very nature, depend upon the pro- perties of the real and of the complex number-fields; such methods may be derived from topology, calculus, convergent series, partial differential equations or analytic function-theory. As examples, one may quote the use of the differ- ential calculus in the proof of the Kronecker-Castelnuovo theorem, of theta- functions in the theory of elliptic curves and abelian varieties, of topology in the proof of the "principle of degeneracy". Let us call "abstract" those methods which, being basically algebraic, are essentially applicable to arbitrary ground- fields; this includes for instance the theory of differentials of the first, second and third kinds (but of course not that of their integrals) and the greater part of the "geometric" proofs of the Italian school. Thus it is plain that, in all cases where an abstract proof is available, it may be expected to yield more than any classical proof for the same result. No one could deny this unless he had made up his mind to ignore fields of non-zero characteristic and was prepared to maintain that a theorem in algebraic geometry which has been proved for the field of complex numbers can always be extended to any field of characteristic 0. There are indeed many cases where this is so; quite often, however, the exten- sion can only be made to algebraically closed fields. As to denying any existence to algebraic geometry of non-zero characteristic, not merely would this, in view of recent developments, amount to denying motion; it would also deprive algebraic geometry of a rich and promising field of possible applications to number-theory, where one cannot do without reduction modulo p.

Serre and Grothendieck describe the contribution of cohomology.
I cannot give a good account of this material in a few words, but I strongly advocate reading these articles which marked the introduction of abstract methods in algebraic geometry in its most fruitful period.