TITLE: Modern algebraic geometry vs. classical algebraic geometry
QUESTION [54 upvotes]: 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?

REPLY [18 votes]: 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 discovery 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.